From a34bbbda1b3e9387db0a48d678920ceea81317e9 Mon Sep 17 00:00:00 2001
From: LehengChen <chenlh@pku.edu.cn>
Date: Sun, 28 Jun 2026 16:32:48 +0800
Subject: [PATCH] fix(riemannian-geometry): revise do Carmo astrolabe OCR text

---
 .../.astrolabe/atoms/049b5985dbae.md          |  8 +--
 .../.astrolabe/atoms/081d380c8cea.md          | 12 ++--
 .../.astrolabe/atoms/093b05290da4.md          | 16 ++---
 .../.astrolabe/atoms/096001bed96a.md          |  4 +-
 .../.astrolabe/atoms/0dc5984b8e92.md          |  7 +-
 .../.astrolabe/atoms/13f6b45547fe.md          |  5 +-
 .../.astrolabe/atoms/167b9abd6797.md          | 13 ++--
 .../.astrolabe/atoms/1a0870f7d0e5.md          |  2 +-
 .../.astrolabe/atoms/1ff5911ba53e.md          |  4 +-
 .../.astrolabe/atoms/2660f4a30491.md          |  4 +-
 .../.astrolabe/atoms/29c2442300b5.md          |  4 +-
 .../.astrolabe/atoms/2b94970df630.md          |  6 +-
 .../.astrolabe/atoms/317503003357.md          |  7 +-
 .../.astrolabe/atoms/3403ca7591a6.md          |  4 +-
 .../.astrolabe/atoms/413b1274c2c2.md          | 10 +--
 .../.astrolabe/atoms/45bcded293f4.md          |  5 +-
 .../.astrolabe/atoms/48f7c97ce541.md          | 10 +--
 .../.astrolabe/atoms/49a53e470b68.md          |  4 +-
 .../.astrolabe/atoms/4c21e0cecff0.md          |  4 +-
 .../.astrolabe/atoms/5665e40876ba.md          |  4 +-
 .../.astrolabe/atoms/58c5cac52c79.md          |  9 +--
 .../.astrolabe/atoms/6136c3f3d16e.md          |  4 +-
 .../.astrolabe/atoms/655ebe66cd00.md          |  4 +-
 .../.astrolabe/atoms/79ff186742ca.md          |  4 +-
 .../.astrolabe/atoms/7eaf6f4a47fc.md          | 13 ++--
 .../.astrolabe/atoms/8286f921d5a2.md          | 10 ++-
 .../.astrolabe/atoms/8a57517fee19.md          | 20 +++---
 .../.astrolabe/atoms/9e5543716f5c.md          |  7 +-
 .../.astrolabe/atoms/a016bbadbb38.md          | 28 +++++---
 .../.astrolabe/atoms/a4eb0713185a.md          |  4 +-
 .../.astrolabe/atoms/a9965ea4095e.md          | 12 ++--
 .../.astrolabe/atoms/ab0d5657b4ac.md          |  4 +-
 .../.astrolabe/atoms/b7028e5e1b4a.md          | 11 ++-
 .../.astrolabe/atoms/b75394389199.md          |  6 +-
 .../.astrolabe/atoms/b9cdcdcd003d.md          | 14 ++--
 .../.astrolabe/atoms/b9e45fa9e39b.md          |  4 +-
 .../.astrolabe/atoms/be5e1aca2bd6.md          |  4 +-
 .../.astrolabe/atoms/c944a1ee39aa.md          |  9 +--
 .../.astrolabe/atoms/cb5eff54c665.md          |  4 +-
 .../.astrolabe/atoms/d483a90d3a7d.md          |  4 +-
 .../.astrolabe/atoms/db5c231a456e.md          |  5 +-
 .../.astrolabe/atoms/e51da6670107.md          |  6 +-
 .../.astrolabe/atoms/e5ec3204f4f6.md          |  4 +-
 .../.astrolabe/atoms/e8d487d40068.md          |  6 +-
 .../.astrolabe/atoms/ed5c99ea7483.md          | 12 ++--
 .../.astrolabe/atoms/ef35a77f11bf.md          | 19 ++---
 .../.astrolabe/atoms/fa0c59ada60c.md          |  8 ++-
 .../.astrolabe/atoms/ff0d95de92e4.md          | 11 +--
 .../.astrolabe/docs-src/00-manifolds.mdx      | 24 ++++---
 .../.astrolabe/docs-src/01-metrics.mdx        | 56 ++++++++-------
 .../.astrolabe/docs-src/02-connections.mdx    | 10 +--
 .../.astrolabe/docs-src/03-geodesics.mdx      | 29 +++++---
 .../.astrolabe/docs-src/04-curvature.mdx      |  2 +-
 .../.astrolabe/docs-src/05-jacobi.mdx         | 11 +--
 .../.astrolabe/docs-src/06-immersions.mdx     |  9 +--
 .../.astrolabe/docs-src/07-hopf-rinow.mdx     |  2 +-
 .../docs-src/08-constant-curvature.mdx        | 70 ++++++++++++-------
 .../.astrolabe/docs-src/09-variations.mdx     |  8 ++-
 .../.astrolabe/docs-src/10-rauch.mdx          | 14 +++-
 .../.astrolabe/docs-src/11-morse.mdx          |  4 +-
 .../docs-src/12-fundamental-group.mdx         | 50 ++++++++-----
 .../.astrolabe/docs-src/13-sphere-theorem.mdx | 39 ++++++++---
 .../.astrolabe/docs/01-metrics.mdx            | 10 +--
 .../.astrolabe/docs/03-geodesics.mdx          |  3 +-
 .../.astrolabe/docs/10-rauch.mdx              |  3 +-
 .../.astrolabe/docs/13-sphere-theorem.mdx     | 20 ++++++
 66 files changed, 463 insertions(+), 280 deletions(-)

diff --git a/projects/riemannian-geometry/.astrolabe/atoms/049b5985dbae.md b/projects/riemannian-geometry/.astrolabe/atoms/049b5985dbae.md
index 226f1b44..1f26363c 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/049b5985dbae.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/049b5985dbae.md
@@ -8,13 +8,13 @@ source: tex
 src: docarmo
 title: covering transformations are translations
 ---
-Let $M$ be a compact Riemannian manifold and $\alpha$ a covering transformation
+Let $M$ be a compact Riemannian manifold and let $\alpha\neq\mathrm{ident}$ be a covering transformation
 of $\tilde{M}$, considered with the covering metric. Then $\alpha$ is a
 translation of $\tilde{M}$.
 
 *Proof.* Let $\tilde{p}\in\tilde{M}$ and let $g\in\pi_1(M;p)$, $p=\pi(\tilde{p})$,
 the element corresponding to $\alpha$ under the isomorphism mentioned in the
-introduction of this Section. We can assume $\alpha\neq\mathrm{ident}$. By
+introduction of this Section. By
 Cartan's Theorem (2.2), there exists a closed geodesic $\gamma$ of $M$ in the
 free homotopy class determined by $g$. Choose a point $q\in\gamma$. Then $\gamma$
 is homotopic to the closed path $\sigma g\sigma^{-1}$, where $\sigma$ is a path
@@ -40,8 +40,8 @@ It follows that if $\tilde{\gamma}(s)$ is a point of the lift of $\gamma$ starti
 from $\tilde{q}$, we have, by uniqueness of lifting,
 
 $$
-\alpha(\tilde{\gamma}(s))=\alpha_{\tilde{q}}(\tilde{\gamma}(s))\in\gamma,
+\alpha(\tilde{\gamma}(s))=\alpha_{\tilde{q}}(\tilde{\gamma}(s))\in\tilde{\gamma},
 $$
 
 which shows that $\tilde{\gamma}$ is invariant by $\alpha$, and proves the
-Proposition. $\blacksquare$
\ No newline at end of file
+Proposition. $\blacksquare$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/081d380c8cea.md b/projects/riemannian-geometry/.astrolabe/atoms/081d380c8cea.md
index 12fdb5d1..9e64ef2e 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/081d380c8cea.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/081d380c8cea.md
@@ -14,10 +14,12 @@ $c(t_0)$, $t_0\in I$ (i.e. $V_0\in T_{c(t_0)}M$). Then there exists a unique
 parallel vector field $V$ along $c$ with $V(t_0)=V_0$; $V(t)$ is called the
 *parallel transport* of $V(t_0)$ along $c$.
 
-*Proof.* By compactness $c([t_0,t_1])$ can be covered by finitely many coordinate
-neighborhoods, and uniqueness makes the local definitions coincide on overlaps,
-so it suffices to prove the theorem when $c(I)$ lies in one coordinate
-neighborhood $\mathbf{x}(U)$. Writing $V=\sum v^j X_j$, parallelism gives
+*Proof.* Suppose first that the theorem is proved when $c(I)$ is contained in a
+local coordinate neighborhood. By compactness, for any $t_1\in I$,
+$c([t_0,t_1])$ can be covered by finitely many coordinate neighborhoods, and
+uniqueness makes the local definitions coincide on overlaps, so it suffices to
+prove the theorem when $c(I)$ lies in one coordinate neighborhood
+$\mathbf{x}(U)$. Writing $V=\sum v^j X_j$, parallelism gives
 $0=\frac{DV}{dt}=\sum_j\frac{dv^j}{dt}X_j+\sum_{i,j}\frac{dx_i}{dt}v^j\nabla_{X_i}X_j$.
 Putting $\nabla_{X_i}X_j=\sum_k\Gamma_{ij}^k X_k$ yields the system of $n$ linear
 ODEs in $v^k(t)$,
@@ -28,4 +30,4 @@ $$
 
 which possesses a unique solution with $v^k(t_0)=v_0^k$. Since the system is
 linear, the solution is defined for all $t\in I$, proving existence and
-uniqueness. $\blacksquare$
\ No newline at end of file
+uniqueness. $\blacksquare$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/093b05290da4.md b/projects/riemannian-geometry/.astrolabe/atoms/093b05290da4.md
index e0806514..4fa360bb 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/093b05290da4.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/093b05290da4.md
@@ -17,7 +17,7 @@ M=B_\rho(p)\cup B_\rho(q),
 $$
 
 where $B_r(p)$ denotes the open geodesic ball of radius $r$ and center $p$, and
-$\rho$ is such that $\pi/2\sqrt\delta<\rho<\pi$.
+$\rho$ is such that $\pi/(2\sqrt\delta)<\rho<\pi$.
 
 *Proof.* By the injectivity-radius estimate of Section 3, $B_\rho(p)$ contains no
 points of $C_m(p)$, so it is diffeomorphic via $\exp$ to a Euclidean ball, and
@@ -32,14 +32,14 @@ $\partial B_\rho(q)$ is path connected, $\partial B_\rho(p)\cap\partial B_\rho(q
 giving $r_o$ with $d(r_o,p)=d(r_o,q)=\rho$. Take $\lambda$ minimizing from $p$ to
 $r_o$; by Berger's Lemma there is $\gamma$ minimizing from $p$ to $q$ with
 $\langle\gamma'(0),\lambda'(0)\rangle\geq0$. Let $s\in\gamma$ with $d(p,s)=\rho$.
-Applying Rauch's Theorem (\entryref{18ec1e38faa8} of Chap. 10), comparing $M^n$ with a
+Applying Rauch's Theorem (\entryref{0a8a561b495c} of Chap. 10), comparing $M^n$ with a
 sphere $S^n$ of curvature $\delta$: since the angle
-$\sphericalangle r_o p s\leq\pi/2$, $d(r_o,s)\leq\pi/2\sqrt\delta$. As
+$\sphericalangle r_o p s\leq\pi/2$, $d(r_o,s)\leq\pi/(2\sqrt\delta)$. As
 $d(r_o,p)=d(r_o,q)=\rho$ and some $s$ has $d(r_o,s)<\rho$, the distance from $r_o$ to
 $\gamma$ is realized by an interior point $s_o$, with the minimizing geodesic from
 $r_o$ to $s_o$ orthogonal to $\gamma$ and
-$d(r_o,\gamma)=d(r_o,s_o)\leq\pi/2\sqrt\delta$. Since $d(p,q)\leq\pi/\sqrt\delta$,
-either $d(p,s_o)\leq\pi/2\sqrt\delta$ or $d(q,s_o)\leq\pi/2\sqrt\delta$; in the
-former case, since $d(r_o,s_o)\leq\pi/2\sqrt\delta$ and
-$\sphericalangle p s_o r_o=\pi/2$, Rauch gives $d(p,r_o)\leq\pi/2\sqrt\delta<\rho$,
-contradicting $d(p,r_o)=\rho$ (the other case is analogous). $\square$
\ No newline at end of file
+$d(r_o,\gamma)=d(r_o,s_o)\leq\pi/(2\sqrt\delta)$. Since $d(p,q)\leq\pi/\sqrt\delta$,
+either $d(p,s_o)\leq\pi/(2\sqrt\delta)$ or $d(q,s_o)\leq\pi/(2\sqrt\delta)$; in the
+former case, since $d(r_o,s_o)\leq\pi/(2\sqrt\delta)$ and
+$\sphericalangle p s_o r_o=\pi/2$, Rauch gives $d(p,r_o)\leq\pi/(2\sqrt\delta)<\rho$,
+contradicting $d(p,r_o)=\rho$ (the other case is analogous). $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/096001bed96a.md b/projects/riemannian-geometry/.astrolabe/atoms/096001bed96a.md
index 96fbb3be..e2a1c8ed 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/096001bed96a.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/096001bed96a.md
@@ -13,7 +13,7 @@ manifold $M^n$ satisfies $0<K\leq 1$, then $i(M)\geq\pi$.
 
 *Proof.* By compactness there are $p,q\in M$ with $q\in C_m(p)$ and $d(p,q)=i(M)$.
 Suppose $d(p,q)<\pi$. If $p$ is conjugate to $q$, then $d(p,q)\geq\pi$ by
-\entryref{be0d62658bc0} of Chap. 10, a contradiction. So $p$ is not conjugate to $q$; by
+\entryref{ba45ce517a2a} of Chap. 10, a contradiction. So $p$ is not conjugate to $q$; by
 Propositions 2.12 and 2.13 there is a closed geodesic $\gamma$ through $p=\gamma(0)$
 and $q$ with $\ell(\gamma)<2\pi$. Parallel transport along $\gamma$ leaves invariant
 a vector $V$ orthogonal to $\gamma$ (Lemma 3.8 of Chap. 9); computing the second
@@ -25,4 +25,4 @@ $q_s=\sigma_s(0)$ to $\gamma_s(0)$ has $\sigma_s(0)\to q$, and an accumulation p
 $w$ of $\sigma_s'(0)$ gives a minimizing geodesic $\sigma(t)=\exp_q tw$ from $q$ to
 $p$. By the first-variation formula $\sigma_s'(0)\perp\gamma_s'$ at $q_s$, so
 $\sigma'(0)\perp\gamma'$ at $q$. This yields three minimizing geodesics from $p$ to
-$q$; since $p$ is not conjugate to $q$, this contradicts \entryref{d8300a62544d}. $\square$
\ No newline at end of file
+$q$; since $p$ is not conjugate to $q$, this contradicts \entryref{d8300a62544d}. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/0dc5984b8e92.md b/projects/riemannian-geometry/.astrolabe/atoms/0dc5984b8e92.md
index bb5c1a12..a70c986a 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/0dc5984b8e92.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/0dc5984b8e92.md
@@ -15,11 +15,12 @@ $$
 (\overline{\nabla}_X B)(Y,Z,\eta)=(\overline{\nabla}_Y B)(X,Z,\eta).
 $$
 
-If, in addition, the codimension of the immersion is $1$, then
+If, in addition, the codimension of the immersion is $1$ and $\eta$ is chosen as a
+local unit normal field, then
 $\nabla_X^\perp\eta=0$, hence
 
 $$
-\overline{\nabla}_X B(Y,Z,\eta)=X\langle S_\eta(Y),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle-\langle S_\eta(Y),\nabla_X Z\rangle=\langle\nabla_X(S_\eta(Y)),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle.
+(\overline{\nabla}_X B)(Y,Z,\eta)=X\langle S_\eta(Y),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle-\langle S_\eta(Y),\nabla_X Z\rangle=\langle\nabla_X(S_\eta(Y)),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle.
 $$
 
 Therefore, in this case, the Codazzi equation can be written
@@ -34,4 +35,4 @@ compatibility equations in the local theory of surfaces (Cf. M. do Carmo [dC 2],
 pp. 235–236). It is possible to state an analogous theorem to the fundamental
 theorem of local surface theory; see K. Tenenblat, "On isometric immersions of
 Riemannian manifolds", Boletim da Soc. Bras. de Mat. vol. 2 (1971), 23–36, and
-H. Jacobowitz, "The Gauss-Codazzi equations", Tensor N.S., 39 (1982), 15–22.
\ No newline at end of file
+H. Jacobowitz, "The Gauss-Codazzi equations", Tensor N.S., 39 (1982), 15–22.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/13f6b45547fe.md b/projects/riemannian-geometry/.astrolabe/atoms/13f6b45547fe.md
index bdec8152..320d72e1 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/13f6b45547fe.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/13f6b45547fe.md
@@ -8,7 +8,8 @@ source: tex
 src: docarmo
 title: the index form
 ---
-It is often convenient to write (5) using
+It is often convenient to write (5) using, on each interval where $V$ is
+differentiable,
 $\frac{d}{dt}\langle V,\frac{DV}{dt}\rangle=\langle V,\frac{D^2 V}{dt^2}\rangle+\langle\frac{DV}{dt},\frac{DV}{dt}\rangle$.
 Taking a geodesic $\gamma:[0,a]\to M$ and a partition
 $0=t_0<t_1<\cdots<t_{k+1}=a$, this gives
@@ -28,4 +29,4 @@ the general case,
 
 $$
 \frac{1}{2}E''(0)=I_a(V,V)-\Big\langle\frac{D}{ds}\frac{\partial f}{\partial s},\gamma'\Big\rangle(0,0)+\Big\langle\frac{D}{ds}\frac{\partial f}{\partial s},\gamma'\Big\rangle(0,a).\qquad(6)
-$$
\ No newline at end of file
+$$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/167b9abd6797.md b/projects/riemannian-geometry/.astrolabe/atoms/167b9abd6797.md
index 42462aa3..e0479c18 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/167b9abd6797.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/167b9abd6797.md
@@ -38,16 +38,18 @@ $$
 Since $K\leq 0$ and $T_c(\tilde{M})$ has zero curvature, we can apply \entryref{0a8a561b495c} (application of Rauch's Theorem) and obtain that
 
 $$
-\ell(\Gamma_C)\leq\ell(\gamma_C)\quad(<,\text{ if }K<0).
+\ell(\Gamma_C)\leq\ell(\gamma_C),
 $$
 
+with strict inequality if $K<0$.
 It follows that
 
 $$
-A^2+B^2-2AB\cos\gamma\leq\ell(\Gamma_C)^2\leq\ell(\gamma_C)^2=C^2\quad(<,\text{ if }K<0),
+A^2+B^2-2AB\cos\gamma\leq\ell(\Gamma_C)^2\leq\ell(\gamma_C)^2=C^2,
 $$
 
-which proves (i).
+where the last inequality is strict if $K<0$.
+This proves (i).
 
 To prove (ii), let us observe that
 
@@ -61,11 +63,12 @@ sides have lengths $A$, $B$ and $C$. Denoting the opposite angles of this triang
 by $\alpha'$, $\beta'$ and $\gamma'$, respectively, we obtain from (i),
 
 $$
-\alpha\leq\alpha',\qquad\beta\leq\beta',\qquad\gamma\leq\gamma'\quad(<,\text{ if }K<0).
+\alpha\leq\alpha',\qquad\beta\leq\beta',\qquad\gamma\leq\gamma'.
 $$
 
+All three inequalities are strict if $K<0$.
 Since $\alpha'+\beta'+\gamma'=\pi$, (ii) follows. $\blacksquare$
 
 From now on, $M$ will denote a complete Riemannian manifold with sectional
 curvature $K<0$. As always, $\pi:\tilde{M}\to M$ denotes the universal covering of
-$M$ with the covering metric. Our goal is to prove the following theorem.
\ No newline at end of file
+$M$ with the covering metric. Our goal is to prove the following theorem.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/1a0870f7d0e5.md b/projects/riemannian-geometry/.astrolabe/atoms/1a0870f7d0e5.md
index 43111524..ad6a5303 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/1a0870f7d0e5.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/1a0870f7d0e5.md
@@ -10,4 +10,4 @@ title: C. Tompkins
 ---
 Suppose that $M$ is compact with zero curvature. If $k<n$, there does not exist
 an isometric immersion $f:M^n\to\mathbb{R}^{n+k}$; in particular, the flat torus
-$T^n$ (see \entryref{5ecedf4a250b}, Chap. 1) cannot be immersed isometrically in $\mathbb{R}^{2n-1}$.
\ No newline at end of file
+$T^n$ (see \entryref{a47005f1c0b9}, Chap. 1) cannot be immersed isometrically in $\mathbb{R}^{2n-1}$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/1ff5911ba53e.md b/projects/riemannian-geometry/.astrolabe/atoms/1ff5911ba53e.md
index 74e4da99..a922d7d4 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/1ff5911ba53e.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/1ff5911ba53e.md
@@ -21,8 +21,10 @@ $s\in(-\varepsilon,\varepsilon)$. If $f$ is differentiable, the variation is sai
 *differentiable*. For each $s\in(-\varepsilon,\varepsilon)$, the parametrized curve
 $f_s:[0,a]\to M$ given by $f_s(t)=f(s,t)$ is called a *curve in the variation*, so
 a variation determines a family $f_s(t)$ of neighboring curves of $f_0(t)=c(t)$.
+The variation is proper exactly when the curves in this family have the same
+initial point $c(0)$ and endpoint $c(a)$.
 The parametrized differentiable curve given by $f_t(s)=f(s,t)$, $t$ fixed, is a
 *transversal curve of the variation*. The velocity vector of a transversal curve
 at $s=0$, defined by $V(t)=\frac{\partial f}{\partial s}(0,t)$, is a (piecewise
 differentiable) vector field along $c(t)$ and is called the *variational field* of
-$f$.
\ No newline at end of file
+$f$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/2660f4a30491.md b/projects/riemannian-geometry/.astrolabe/atoms/2660f4a30491.md
index c29b2a56..1de9b370 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/2660f4a30491.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/2660f4a30491.md
@@ -16,10 +16,10 @@ $0<K_{\min}\leq K\leq K_{\max}$, then:
 
 *Proof.* By Bonnet–Myers (Theor. 3.1 of Chap. 9), $M$ is compact. Since $T_1M$ is
 compact, \entryref{31ea0ff2d398} gives $p\in M$ with $d(p,C_m(p))=\inf_{r\in M}d(r,C_m(r))$,
-and $C_m(p)$ compact gives $q\in C_m(p)$ assuming this distance. If $q$ is conjugate
+and $C_m(p)$ compact gives $q\in C_m(p)$ realizing this distance. If $q$ is conjugate
 to $p$, then $d(p,q)\geq\pi/\sqrt{K_{\max}}$ by \entryref{ba45ce517a2a} — case
 (a). If $q$ is not conjugate to $p$, by \entryref{d8300a62544d} there are two minimizing
 geodesics $\mu,\sigma$ from $p$ to $q$ with $\mu'(\ell)=-\sigma'(\ell)$,
 $\ell=d(p,q)$. Since $q\in C_m(p)$ we have $p\in C_m(q)$ and $p$ realizes the
 distance from $q$ to $C_m(q)$, so $\mu'(0)=-\sigma'(0)$; thus $\mu$ and $\sigma$ form
-a closed geodesic $\gamma$ satisfying (b). $\square$
\ No newline at end of file
+a closed geodesic $\gamma$ satisfying (b). $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/29c2442300b5.md b/projects/riemannian-geometry/.astrolabe/atoms/29c2442300b5.md
index cc22fccc..c288dff5 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/29c2442300b5.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/29c2442300b5.md
@@ -29,9 +29,9 @@ with $\varepsilon_j'\leq\varepsilon_j$ gives
 $\exp_p(t_0+\varepsilon_j)\gamma'(0)=\exp_p(t_0+\varepsilon_j')\sigma_j'(0)$, hence
 $\gamma'(0)=\sigma_j'(0)$, contradicting the definition of $t_0$. Conversely, if
 (a) holds, since a geodesic does not minimize after the first conjugate point
-(\entryref{31ea0ff2d398} to the Index Theorem, Chap. 11), the cut point occurs at
+(\entryref{be5e1aca2bd6} to the Index Theorem, Chap. 11), the cut point occurs at
 $\gamma(\tilde t)$, $\tilde t\leq t_0$. If (b) holds, joining $\sigma(t_0-\varepsilon)$
 to $\gamma(t_0+\varepsilon)$ by the unique minimizing geodesic $\tau$ inside a
 totally normal neighborhood of $\gamma(t_0)$ yields a curve of length strictly less
 than $t_0+\varepsilon$, so again the cut point occurs at $\gamma(\tilde t)$,
-$\tilde t\leq t_0$. $\square$
\ No newline at end of file
+$\tilde t\leq t_0$. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/2b94970df630.md b/projects/riemannian-geometry/.astrolabe/atoms/2b94970df630.md
index 64b3cb56..a1e736e3 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/2b94970df630.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/2b94970df630.md
@@ -8,9 +8,9 @@ source: tex
 src: docarmo
 title: Corollary 2.5
 ---
-Let $\gamma:[0,a]\to M$ be a geodesic. Then a Jacobi field $J$ along $\gamma$ with
-$J(0)=0$ is given by
+Let $\gamma:[0,a]\to M$ be a geodesic and put $p=\gamma(0)$. Then a Jacobi field
+$J$ along $\gamma$ with $J(0)=0$ is given by
 
 $$
 J(t)=(d\exp_p)_{t\gamma'(0)}(tJ'(0)),\quad t\in[0,a].
-$$
\ No newline at end of file
+$$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/317503003357.md b/projects/riemannian-geometry/.astrolabe/atoms/317503003357.md
index 85be49fb..5c128c2e 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/317503003357.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/317503003357.md
@@ -13,7 +13,10 @@ $q\in M$ to the geodesic sphere $S_r(p)$ of radius $r<c$ stays outside the
 geodesic ball $B_r(p)$ in some neighborhood of $q$.
 *Proof.* Let $W$ be a totally normal neighborhood of $p$; by the lemma of
 homogeneity restrict to unit-speed geodesics, so to the unit bundle
-$T_1W=\{(q,v);\ q\in W,\ v\in T_qM,\ |v|=1\}$. Let
+$T_1W=\{(q,v);\ q\in W,\ v\in T_qM,\ |v|=1\}$. After shrinking $W$, assume that
+$\overline{W}$ is compactly contained in a normal neighborhood $U$ of $p$; since
+$T_1\overline{W}$ is compact, decrease $I$ so that
+$\gamma(I\times T_1W)\subset U$. Let
 $\gamma:I\times T_1W\to M$, $I=(-\varepsilon,\varepsilon)$, be the differentiable
 map with $t\mapsto\gamma(t,q,v)$ the unit-speed geodesic through $q$ with velocity
 $v$ at $t=0$. Set $u(t,q,v)=\exp_p^{-1}(\gamma(t,q,v))$ and
@@ -33,4 +36,4 @@ $\frac{\partial^2 F}{\partial t^2}(0,p,v)=2|v|^2=2>0$. Hence there is a
 neighborhood $V\subset W$ of $p$ with $\frac{\partial^2 F}{\partial t^2}(0,q,v)>0$
 for all $q\in V$, $|v|=1$. Take $c>0$ with $\exp_p B_c(0)\subset V$: any geodesic
 in $B_c(p)$ tangent to $S_r(p)$, $r<c$, has a strict local minimum of $F$ at
-$(0,q,v)$, so in a neighborhood of $q$ its points stay outside $B_r(p)$. $\square$
\ No newline at end of file
+$(0,q,v)$, so in a neighborhood of $q$ its points stay outside $B_r(p)$. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/3403ca7591a6.md b/projects/riemannian-geometry/.astrolabe/atoms/3403ca7591a6.md
index 41a74cd5..cea0d425 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/3403ca7591a6.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/3403ca7591a6.md
@@ -14,6 +14,6 @@ orthogonal complement of $\gamma'(0)$. We say that $\gamma$ is *focal point free
 at $(0,a]$ if there exists some $\varepsilon>0$ such that $\gamma$ has no focal
 points relative to the submanifold $\Sigma_\varepsilon=\exp_{\gamma(0)}(B_\varepsilon(0))$.
 
-Observe that since $\Sigma_\varepsilon$ is geodesic at $p$, any Jacobi field $J$
+Observe that since $\Sigma_\varepsilon$ is geodesic at $\gamma(0)$, any Jacobi field $J$
 along $\gamma$, with $J(0)\neq 0$ and $J'(0)=0$, automatically satisfies
-$S_{\gamma'(0)}(J(0))=0$.
\ No newline at end of file
+$S_{\gamma'(0)}(J(0))=0$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/413b1274c2c2.md b/projects/riemannian-geometry/.astrolabe/atoms/413b1274c2c2.md
index 880bd473..0971da97 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/413b1274c2c2.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/413b1274c2c2.md
@@ -64,11 +64,12 @@ $$
 $$
 
 which we write as $1/\lambda=\rho=a_1|p-p_0|^2+k_1$ with $a_1=\sigma/2$,
-$k_1=\text{const.}$, $p_0\in\mathbb{R}^n$. Taking the inversion
+$k_1=\text{const.}$, $p_0\in\mathbb{R}^n$. The proof in this case will be complete
+once we show $k_1=0$: if $k_1=0$, taking the inversion
 $g(p)=\frac{p-p_0}{|p-p_0|^2}+p_0$ and $h=g\circ f^{-1}$, $h$ is conformal with
 coefficient $a_1$, hence an isometry followed by a dilatation; thus
 $f=h^{-1}\circ g$ is an inversion followed by a dilatation, followed by an
-isometry. To show $k_1=0$: applying the argument to $f^{-1}$,
+isometry. To show $k_1=0$, apply the argument to $f^{-1}$:
 $\lambda=a_2|f(p)-q_0|^2+k_2$, hence
 
 $$
@@ -84,7 +85,8 @@ $$
 $$
 
 If $k_1\neq 0$, the left side is a transcendental function of $|p(s_0)-p_0|$, while
-$(11)$ implies it is algebraic. This contradiction shows $k_1=0$.
+$(11)$ implies it is algebraic. This contradiction shows $k_1=0$, completing the
+case $\sigma\neq0$.
 
 *Case $\sigma=0$.* Here $\rho=1/\lambda=\sum a_i x_i+c_1=A_1(x)+c_1$. Applying the
 initial argument to $f^{-1}$,
@@ -103,4 +105,4 @@ $$
 
 which contradicts $(11')$ unless $A_1(p(s))\equiv 0$. We conclude that if
 $\sigma=0$, $\lambda=\text{const.}$; then lengths of tangent vectors are multiplied
-by a constant $\lambda$ and $f$ is an isometry followed by a dilatation. $\square$
\ No newline at end of file
+by a constant $\lambda$ and $f$ is an isometry followed by a dilatation. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/45bcded293f4.md b/projects/riemannian-geometry/.astrolabe/atoms/45bcded293f4.md
index 779200d6..e5d6659f 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/45bcded293f4.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/45bcded293f4.md
@@ -10,7 +10,8 @@ title: the Lobatchevski plane
 ---
 Let $G=\{(x,y)\in\mathbb{R}^2;\ y>0\}$ with metric $g_{11}=g_{22}=\frac{1}{y^2}$,
 $g_{12}=g_{21}=0$. The $y$-axis segment $\gamma(t)=(0,t)$, $a\le t\le b$ ($a>0$),
-is a geodesic: for any arc $c(t)=(x(t),y(t))$ with $c(a)=(0,a)$, $c(b)=(0,b)$,
+is the image of a geodesic: for any arc $c(t)=(x(t),y(t))$ with $c(a)=(0,a)$,
+$c(b)=(0,b)$,
 
 $$
 \ell(c)=\int_a^b\sqrt{(x')^2+(y')^2}\,\frac{dt}{y}\ge\int_a^b\frac{|y'|}{y}\,dt\ge\int_a^b\frac{dy}{y}=\ell(\gamma),
@@ -21,4 +22,4 @@ isometries $z\mapsto\frac{az+b}{cz+d}$, $z=x+iy$, $ad-bc=1$ (cf. Exercise 4 of
 Chap. 1) carry the $y$-axis into upper semicircles or rays $x=x_0$, $y>0$, which
 are therefore geodesics; these are all the geodesics, since through each $p\in G$
 and each direction there passes such a circle centered on the $x$-axis (a normal
-direction giving a vertical ray).
\ No newline at end of file
+direction giving a vertical ray).
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/48f7c97ce541.md b/projects/riemannian-geometry/.astrolabe/atoms/48f7c97ce541.md
index cd5311cd..76f4af52 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/48f7c97ce541.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/48f7c97ce541.md
@@ -42,9 +42,11 @@ $$
 differential $\mathrm{Ad}(a)=dR_{a^{-1}}dL_a:\mathcal{G}\to\mathcal{G}$, so
 $\mathrm{Ad}(a)Y=dR_{a^{-1}}Y$. Let $x_t$ be the flow of $X$. From \entryref{453ecde8947b}, $[Y,X]=\lim_{t\to0}\frac1t(dx_t(Y)-Y)$. Since $X$ is left invariant,
 $L_y\circ x_t=x_t\circ L_y$, giving $x_t(y)=R_{x_t(e)}(y)$, hence $dx_t=dR_{x_t(e)}$
-and $[Y,X]=\lim_{t\to0}\frac1t(\mathrm{Ad}(x_t^{-1}(e))Y-Y)$. With $\langle\ ,\ \rangle$
+and $[Y,X]=\lim_{t\to0}\frac1t(\mathrm{Ad}((x_t(e))^{-1})Y-Y)$. With $\langle\ ,\ \rangle$
 bi-invariant, $\langle U,V\rangle=\langle dR_{x_t(e)}U,dR_{x_t(e)}V\rangle$;
 differentiating in $t$ at $t=0$ gives $0=\langle[U,X],V\rangle+\langle U,[V,X]\rangle$,
-which is (3). The relation (3) characterizes bi-invariant metrics: a positive
-bilinear form on $\mathcal{G}$ satisfying (3) yields, via (2), a bi-invariant
-metric. $\blacksquare$
\ No newline at end of file
+which is (3). $\blacksquare$
+
+The relation (3) characterizes bi-invariant metrics: a positive bilinear form on
+$\mathcal{G}$ satisfying (3) yields, via (2), a bi-invariant metric. This last
+fact is not proved here.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/49a53e470b68.md b/projects/riemannian-geometry/.astrolabe/atoms/49a53e470b68.md
index 890cf0bc..c6ec86fb 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/49a53e470b68.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/49a53e470b68.md
@@ -11,10 +11,10 @@ title: convex neighborhoods
 For any $p\in M$ there exists $\beta>0$ such that the geodesic ball $B_\beta(p)$ is
 strongly convex.
 *Proof.* Let $c$ be as in \entryref{317503003357}. Choose $\delta>0$ and $W$ as in \entryref{f29bd1e6ebb3}
-with $\delta<\frac{c}{2}$, and take $\beta<\delta$ with $B_\beta(p)\subset W$. Let
+with $\delta<\frac{c}{2}$, and take $\beta<\delta$ with $\overline{B_\beta(p)}\subset W$. Let
 $q_1,q_2\in\overline{B_\beta(p)}$ and let $\gamma$ be the unique geodesic of length
 $<2\delta<c$ joining them; then $\gamma\subset B_c(p)$. If the interior of
 $\gamma$ is not contained in $B_\beta(p)$, there is an interior point $m$ where the
 maximum distance $r$ from $p$ to $\gamma$ is attained; near $m$ the points of
 $\gamma$ remain in $\overline{B_r(p)}$. Since $m\in B_c(p)$ this contradicts
-\entryref{317503003357}, proving the proposition. $\square$
\ No newline at end of file
+\entryref{317503003357}, proving the proposition. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/4c21e0cecff0.md b/projects/riemannian-geometry/.astrolabe/atoms/4c21e0cecff0.md
index 9603410b..71db0ab7 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/4c21e0cecff0.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/4c21e0cecff0.md
@@ -8,6 +8,6 @@ source: tex
 src: docarmo
 title: Remark 3.8
 ---
-The torus $T^{n+1}$ with the flat metric (Cf. \entryref{5ecedf4a250b}, Chap. 1) contains the
+The torus $T^{n+1}$ with the flat metric (Cf. \entryref{a47005f1c0b9}, Chap. 1) contains the
 torus $T^n$ as a totally geodesic submanifold. This shows that the hypothesis
-that $\overline{M}$ be simply connected in \entryref{d483a90d3a7d} is necessary.
\ No newline at end of file
+that $\overline{M}$ be simply connected in \entryref{d483a90d3a7d} is necessary.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/5665e40876ba.md b/projects/riemannian-geometry/.astrolabe/atoms/5665e40876ba.md
index a80790cd..eb6ec3ca 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/5665e40876ba.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/5665e40876ba.md
@@ -9,10 +9,10 @@ src: docarmo
 title: Gauss formula for a hypersurface
 ---
 In the case of a hypersurface $f:M^n\to\overline{M}^{n+1}$, let $p\in M$,
-$\eta\in(T_pM)^\perp$, and $\{e_1,\dots,e_n\}$ an orthonormal basis of $T_pM$ in
+$\eta\in(T_pM)^\perp$, $|\eta|=1$, and $\{e_1,\dots,e_n\}$ an orthonormal basis of $T_pM$ in
 which $S_\eta=S$ is diagonal, $S(e_i)=\lambda_ie_i$. Then $H(e_i,e_i)=\lambda_i$
 and $H(e_i,e_j)=0$ if $i\neq j$, so (1) becomes
 
 $$
 K(e_i,e_j)-\overline{K}(e_i,e_j)=\lambda_i\lambda_j.\qquad(4)
-$$
\ No newline at end of file
+$$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/58c5cac52c79.md b/projects/riemannian-geometry/.astrolabe/atoms/58c5cac52c79.md
index 032c6af3..4fa689a0 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/58c5cac52c79.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/58c5cac52c79.md
@@ -10,8 +10,9 @@ title: Remark 3.11
 ---
 Riemannian manifolds of non-positive curvature form a substantial topic in
 Riemannian Geometry, which we have barely touched. To the reader interested in
-some of the recent developments, we recommend the survey by P. Eberlein,
+some of the recent developments, we recommend the excellent survey by P. Eberlein,
 "Structure of manifolds of nonpositive curvature", in *Global Geometry and Global
-Analysis 1984*, Proceedings, Berlin, Lecture Notes in Math. 1156, Springer Verlag,
-1985, pp. 86–153. Other relationships between the geometry of a manifold $M$ of
-non-positive curvature and $\pi_1(M)$ are described in Section 7 of this survey.
\ No newline at end of file
+Analysis 1984*, Proceedings, Berlin, edited by D. Ferus, R. Gardner, S. Helgason
+and U. Simon, Lecture Notes in Math. 1156, Springer Verlag, 1985, pp. 86–153.
+Other relationships between the geometry of a manifold $M$ of
+non-positive curvature and $\pi_1(M)$ are described in Section 7 of this survey.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/6136c3f3d16e.md b/projects/riemannian-geometry/.astrolabe/atoms/6136c3f3d16e.md
index 54b2abd5..4f82dad6 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/6136c3f3d16e.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/6136c3f3d16e.md
@@ -20,7 +20,7 @@ i(t+\varepsilon)\leq n(j-1)-\{n(j-1)-i(t)-d\}=i(t)+d.
 $$
 
 To prove the inequality in the other direction, it is necessary to use the Index
-Lemma (\entryref{a8a28b2c19fb} of Chap. 10). Let $V\in S_j$, with $V(t_{j-1})\neq 0$, and
+Lemma (\entryref{7a9c6afeabeb} of Chap. 10). Let $V\in S_j$, with $V(t_{j-1})\neq 0$, and
 denote by $V_{t_o}$ the "broken" Jacobi field which coincides with $V(t_i)$ at
 $t_i$, $i=1,\dots,j-1$, and which vanishes at the point $t_o\in(t_{j-1},t_j)$. We
 claim that
@@ -61,4 +61,4 @@ The information which we obtained about $i(t)$ allows us to describe $i(t)$ as a
 function which is zero in a neighborhood of the origin, is continuous on the left
 and has "step" type discontinuities at the conjugate points to $\gamma(0)$, the
 step being exactly equal to the multiplicity of the conjugate point (see Fig. 2).
-But this is precisely the statement of the Index Theorem. $\square$
\ No newline at end of file
+But this is precisely the statement of the Index Theorem. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/655ebe66cd00.md b/projects/riemannian-geometry/.astrolabe/atoms/655ebe66cd00.md
index 230509f2..ac51f7da 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/655ebe66cd00.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/655ebe66cd00.md
@@ -10,7 +10,7 @@ title: no conjugate points when $K\le 0$
 ---
 Let $M$ be a complete Riemannian manifold with $K(p,\sigma)\le 0$ for all $p\in M$
 and for all $\sigma\subset T_pM$. Then for all $p\in M$ the conjugate locus
-$C(p)=\phi$; in particular the exponential map $\exp_p:T_pM\to M$ is a local
+$C(p)=\varnothing$; in particular the exponential map $\exp_p:T_pM\to M$ is a local
 diffeomorphism.
 *Proof.* Let $J$ be a non-trivial (that is, not identically zero) Jacobi field
 along a geodesic $\gamma:[0,\infty)\to M$, where $\gamma(0)=p$ and $J(0)=0$. Then
@@ -36,4 +36,4 @@ It follows that for all $t>0$, $\langle J,J\rangle(t)>0$, and $\gamma(t)$ is not
 conjugate to $\gamma(0)$ along $\gamma$. $\square$
 
 The crucial point in the proof of the Hadamard theorem is given in the lemma
-below, which is of independent interest.
\ No newline at end of file
+below, which is of independent interest.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/79ff186742ca.md b/projects/riemannian-geometry/.astrolabe/atoms/79ff186742ca.md
index 3a724e2c..71942dd9 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/79ff186742ca.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/79ff186742ca.md
@@ -9,7 +9,7 @@ src: docarmo
 title: minimal immersion; mean curvature vector
 ---
 An immersion $f:M\to\overline{M}$ is called *minimal* if for every $p\in M$ and
-every $\eta\in(T_pM)^\perp$ the trace of $S_\eta=0$. Choosing an orthonormal frame
+every $\eta\in(T_pM)^\perp$ the trace of $S_\eta$ is zero. Choosing an orthonormal frame
 $E_1,\dots,E_m$ of vectors in $\mathcal{X}(U)^\perp$, where $U$ is a neighborhood
 of $p$ in which $f$ is an embedding, we write at $p$
 
@@ -26,4 +26,4 @@ $$
 where $S_i=S_{E_i}$, does not depend on the chosen frame; it is the *mean
 curvature vector* of $f$. Thus $f$ is minimal iff $H(p)=0$ for all $p\in M$. Such
 immersions minimize volume in the induced metric, in the same way that geodesics
-minimize arc length.
\ No newline at end of file
+minimize arc length.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/7eaf6f4a47fc.md b/projects/riemannian-geometry/.astrolabe/atoms/7eaf6f4a47fc.md
index 8ea6b30f..a412d868 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/7eaf6f4a47fc.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/7eaf6f4a47fc.md
@@ -15,8 +15,11 @@ $\pi_1(x)=\big(\frac{x_1}{1-x_{n+1}},\dots,\frac{x_n}{1-x_{n+1}}\big)$, and
 $\pi_2:S^n-\{S\}\to\mathbb{R}^n$ (from the south pole) are differentiable
 injections onto $x_{n+1}=0$. The parametrizations
 $(\mathbb{R}^n,\pi_1^{-1}),(\mathbb{R}^n,\pi_2^{-1})$ cover $S^n$ with change of
-coordinates $y_j'=\frac{y_j}{\sum_{i=1}^n y_i^2}$, differentiable, and
-$\pi_1^{-1}(\mathbb{R}^n)\cap\pi_2^{-1}(\mathbb{R}^n)=S^n-\{N\cup S\}$ connected;
-by \entryref{6bc07ca778f4}, $S^n$ is orientable. The antipodal map $A:S^n\to S^n$, $A(p)=-p$,
-is a diffeomorphism ($A^2=\mathrm{id}$); it reverses the orientation of $S^n$ when
-$n$ is even and preserves it when $n$ is odd.
\ No newline at end of file
+coordinates $y_j'=\frac{y_j}{\sum_{i=1}^n y_i^2}$, differentiable. For $n\geq2$,
+$\pi_1^{-1}(\mathbb{R}^n)\cap\pi_2^{-1}(\mathbb{R}^n)=S^n-\{N,S\}$ is connected,
+so by \entryref{6bc07ca778f4}, $S^n$ is orientable. For $n=1$, the same atlas is
+orientable directly: on the two components of $S^1-\{N,S\}$, the change of
+coordinates is $y'=1/y$ with derivative $-1/y^2<0$, and changing the sign of one
+coordinate makes both transition derivatives positive. The antipodal map
+$A:S^n\to S^n$, $A(p)=-p$, is a diffeomorphism ($A^2=\mathrm{id}$); it reverses the
+orientation of $S^n$ when $n$ is even and preserves it when $n$ is odd.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/8286f921d5a2.md b/projects/riemannian-geometry/.astrolabe/atoms/8286f921d5a2.md
index 5c3b8bf3..ba2a7b5b 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/8286f921d5a2.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/8286f921d5a2.md
@@ -13,7 +13,7 @@ geodesics, with the same velocity, and let $J$ and $\tilde J$ be Jacobi fields
 along $\gamma$ and $\tilde\gamma$, such that
 
 $$
-0=J'(0)=\tilde J'(0),\quad\langle J(0),\gamma'(0)\rangle=\langle J(0),\tilde\gamma'(0)\rangle,\quad|J(0)|=|\tilde J(0)|.
+0=J'(0)=\tilde J'(0),\quad\langle J(0),\gamma'(0)\rangle=\langle \tilde J(0),\tilde\gamma'(0)\rangle,\quad|J(0)|=|\tilde J(0)|.
 $$
 
 Assume that $\tilde\gamma$ is focal point free on $(0,a]$ and that, for all $t$
@@ -33,6 +33,12 @@ modifications: $\frac{|J|^2}{|\tilde J|^2}$ is well-defined, since $\tilde\gamma
 is focal point free, and $I_{t_o}(\phi(U),\phi(U))\geq I_{t_o}(\tilde U,\tilde U)$
 by \entryref{cdfac76c48af}. $\blacksquare$
 
+Other useful extensions of the Rauch comparison theorem can be found in F. Warner,
+"Extensions of the Rauch comparison theorem to submanifolds", Trans. A.M.S. 122
+(1966), 341-356, and in E. Heintze and H. Karcher, "A general comparison theorem
+with applications to volume estimates for submanifolds", Ann. Sci. Ecole Norm.
+Sup. 11 (1978), 451-470.
+
 Rauch's theorem admits an extremely important global generalization, called the
 Theorem of Toponogov. One of its versions: let $M$ be a Riemannian manifold which
 is complete with sectional curvature $K\geq H$. Let $\gamma_1$ and $\gamma_2$ be
@@ -50,4 +56,4 @@ $$
 
 A proof can be found in Cheeger and Ebin [CE]. The Theorem of Toponogov is an
 essential tool for the study of the relation between topology and curvature; one
-of the culmination points, the sphere theorem, is presented in Chapter 13.
\ No newline at end of file
+of the culmination points, the sphere theorem, is presented in Chapter 13.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/8a57517fee19.md b/projects/riemannian-geometry/.astrolabe/atoms/8a57517fee19.md
index 5e86874d..1d9b6eae 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/8a57517fee19.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/8a57517fee19.md
@@ -18,16 +18,17 @@ onto $H^n$. Conversely, let $f:H^n\to H^n$ be a conformal transformation of $H^n
 onto $H^n$ and let $e_1,\dots,e_n$ be an orthonormal basis (in $g_{ij}$) at
 $p\in H^n$. Since $g_{ij}=\delta_{ij}/x_n^2$ and $f$ is conformal, there exists
 $\lambda^2>0$ with $\langle df_p(e_i),df_p(e_j)\rangle=\lambda^2\delta_{ij}$, so
-$\{df_p(e_i)/\lambda\}$ is orthonormal at $f(p)$. By \entryref{08fee88dd521} of Cartan's
-Theorem, there exists an isometry $g$ of $H^n$ taking $p$ to $f(p)$ with
-$dg(e_i)=df(e_i)/\lambda$. By the first part, $g$ is conformal. Hence
+$\{df_p(e_i)/\lambda\}$ is orthonormal at $f(p)$. \entryref{08fee88dd521} of Cartan's
+Theorem, together with the global construction after Theorem 2.1 (the exponential
+maps of $H^n$ are global diffeomorphisms), gives an isometry $g$ of $H^n$ taking
+$p$ to $f(p)$ with $dg(e_i)=df(e_i)/\lambda$. By the first part, $g$ is conformal. Hence
 $h=g^{-1}\circ f$ is the restriction to $H^n$ of a conformal map of $\mathbb{R}^n$
 taking $H^n$ onto $H^n$, leaving $p$ fixed and satisfying $dh_p=\lambda I$. It
 remains to show $h=\text{identity}$.
 
 Let $P$ be a hyperplane through $p$. By Liouville's theorem, $h(P)$ is a hyperplane
 or a sphere through $p$; since $dh_p$ is a multiple of the identity, $P$ and $h(P)$
-are tangent at $P$. Since $h$ takes $\partial H^n$ into itself and is conformal,
+are tangent at $p$. Since $h$ takes $\partial H^n$ into itself and is conformal,
 the angle of $P$ with $\partial H^n$ equals the angle of $h(P)$ with $\partial H^n$.
 We claim $h(P)=P$. Consider a line $r_1$ through $p$ perpendicular to
 $\partial H^n$, $q_1=r_1\cap\partial H^n$. Since $h(r_1)$ is a circle or a line
@@ -43,15 +44,18 @@ onto $H^n$, $h$ is the identity.
 For $n=2$, a simple calculation shows the conformal transformations of the form
 
 $$
-f(z)=\frac{az+b}{cz+d},\quad z\in H^2\subset\mathbb{C},\ a,b,c,d\in\mathbb{R},\ ad-bc\neq 0\qquad(12)
+f(z)=\frac{az+b}{cz+d},\quad z\in H^2\subset\mathbb{C},\ a,b,c,d\in\mathbb{R},\ ad-bc>0\qquad(12)
 $$
 
 (which map $H^2$ onto $H^2$) are isometries of $H^2$ with the metric $g_{ij}$.
 Moreover, for a fixed point $p_0\in H^2$ and unit vector $v_0$ at $p_0$, there
 exists a transformation $(12)$ taking an arbitrary $(p,v)$ to $(p_0,v_0)$ ($p$ and
 $v$ are determined by three parameters, the number of parameters of $f$). Since
-there is a unique isometry of $H^2$ taking $(p,v)$ to $(p_0,v_0)$, all isometries
-of $H^2$ are of the form $(12)$. $\square$
+there is a unique orientation-preserving isometry of $H^2$ taking $(p,v)$ to
+$(p_0,v_0)$, all orientation-preserving isometries of $H^2$ are of the form (12).
+Composing these with $z\mapsto-\bar z$ gives the orientation-reversing
+anti-holomorphic isometries, so together they give all isometries of $H^2$.
+$\square$
 
 To conclude this section, we identify some important hypersurfaces of $H^n$. The
 intersection with $H^n$ of hyperplanes of $\mathbb{R}^n$ orthogonal to $\partial H^n$,
@@ -62,4 +66,4 @@ $B^n=\{p\in\mathbb{R}^n;\ |p|<2\}$ with the metric
 
 $$
 h_{ij}(p)=\frac{\delta_{ij}}{(1-\tfrac14|p|^2)^2}.
-$$
\ No newline at end of file
+$$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/9e5543716f5c.md b/projects/riemannian-geometry/.astrolabe/atoms/9e5543716f5c.md
index 10be4fa6..03443bce 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/9e5543716f5c.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/9e5543716f5c.md
@@ -8,8 +8,9 @@ source: tex
 src: docarmo
 title: special cases of Example 4.8
 ---
-**4.9 (a) (the $k$-torus).** $G$ = group of "integral" translations of
-$\mathbb{R}^k$, $G(x_1,\dots,x_k)=(x_1+n_1,\dots,x_k+n_k)$, $n_i\in\mathbb{Z}$.
+**4.9 (a) (the $k$-torus).** Let $G$ be the group of "integral" translations of
+$\mathbb{R}^k$: for $(n_1,\dots,n_k)\in\mathbb{Z}^k$, the corresponding
+translation sends $(x_1,\dots,x_k)$ to $(x_1+n_1,\dots,x_k+n_k)$.
 This is a properly discontinuous action; $\mathbb{R}^k/G$ (with the structure of
 \entryref{32ee361d33e8}) is the *$k$-torus $T^k$*. For $k=2$, $T^2$ is diffeomorphic to the
 torus of revolution in $\mathbb{R}^3$, the inverse image of $0$ of
@@ -21,4 +22,4 @@ $\{A,\mathrm{Id}\}$ acts properly discontinuously on $S$; $S/G$ carries the
 structure of \entryref{32ee361d33e8}. When $S$ is the torus of revolution $T^2$, $S/G=K$ is
 *the Klein bottle*; when $S=C=\{(x,y,z); x^2+y^2=1,\,-1<z<1\}$ (a right circular
 cylinder), $S/G$ is the *Möbius band*. By Exercise 9 the Klein bottle and Möbius
-band are non-orientable.
\ No newline at end of file
+band are non-orientable.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/a016bbadbb38.md b/projects/riemannian-geometry/.astrolabe/atoms/a016bbadbb38.md
index 2217e06a..c6973234 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/a016bbadbb38.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/a016bbadbb38.md
@@ -12,17 +12,20 @@ If $M$ is compact, $K<0$, and $H$ is a solvable subgroup of $\pi_1(M)$,
 $H\neq\{e\}$, then $H$ is infinite cyclic. In addition, $\pi_1(M)$ does not have a
 cyclic subgroup of finite index.
 
-*Proof.* Since $H$ is solvable, there exists a finite sequence of subgroups
+*Proof.* Since $H$ is solvable, choose a finite sequence of subgroups with $k$
+minimal:
 
 $$
-H=H_o\supset H_1\supset\cdots\supset H_{k-1}\supset H_k=\{e\}
+H=H_0\supset H_1\supset\cdots\supset H_{k-1}\supset H_k=\{e\}
 $$
 
 such that $H_{i+1}$ is normal in $H_i$ and $H_i/H_{i+1}$ is abelian. Then
-$H_{k-1}$ is abelian, hence, by Preissman's Theorem (3.2), infinite cyclic.
+$H_{k-1}\neq\{e\}$ and $H_{k-1}$ is abelian, hence, by Preissman's Theorem (3.2),
+infinite cyclic. If $k=1$, this proves the first assertion. Assume $k\geq2$.
 
 Let $g\in\pi_1(M)$ be a generator of $H_{k-1}$ and let $\tilde{\gamma}$ be the
-geodesic in $\tilde{M}$ invariant under $g$. Let $a\in H_{k-2}$ and $b\in H_{k-1}$.
+geodesic in $\tilde{M}$ invariant under $g$. Let $a\in H_{k-2}$, and choose a
+nonidentity element $b\in H_{k-1}$, for instance $b=g$.
 Since $a^{-1}b^{-1}ab\in H_{k-1}$, we have, for some integer $n$,
 
 $$
@@ -45,16 +48,21 @@ Repeating the argument above a finite number of times, we conclude that $H$ is
 infinite cyclic, which proves the first statement of the theorem.
 
 To prove the final assertion, suppose that there exists a subgroup
-$H\subset\pi_1(M)$, cyclic and of finite index. Let $g$ be a generator of $H$ and
-let $\tilde{\gamma}$ be the geodesic of $\tilde{M}$ invariant under $g$. Let
-$a\in\pi_1(M)-H$. Since $H$ has finite index, there exist integers $m$ and $n$
-such that $a^n=g^m$. Therefore
+$H\subset\pi_1(M)$, cyclic and of finite index. If $H=\{e\}$, then $\pi_1(M)$ is
+finite; this contradicts Preissman's Theorem (3.2) unless $\pi_1(M)=\{e\}$, which
+contradicts \entryref{f8456cb22b9d}. Thus $H=\langle g\rangle$ with $g\neq e$. If
+$H=\pi_1(M)$, then $\pi_1(M)$ is abelian, again contradicting \entryref{f8456cb22b9d}. Hence
+$H$ is proper. Let $\tilde{\gamma}$ be the geodesic of $\tilde{M}$ invariant under
+$g$, and let $a\in\pi_1(M)-H$. Since $H$ has finite index, choose $n>0$ with
+$a^n\in H$. Applying Preissman's Theorem to the subgroup generated by $a$, we have
+$a^n\neq e$, so $a^n=g^m$ for some $m\neq0$. Therefore
 
 $$
 a^n(\tilde{\gamma})=g^m(\tilde{\gamma})=\tilde{\gamma}.
 $$
 
-By uniqueness, $a(\tilde{\gamma})=\tilde{\gamma}$, for all $a\in\pi_1(M)-H$. It
+The nonidentity transformation $a^n$ also leaves $a(\tilde{\gamma})$ invariant, so
+by uniqueness, $a(\tilde{\gamma})=\tilde{\gamma}$, for all $a\in\pi_1(M)-H$. It
 follows that every element of $\pi_1(M)$ leaves invariant the geodesic
 $\tilde{\gamma}$. By \entryref{46baf7fab729}, $\pi_1(M)$ is infinite cyclic, which contradicts
-\entryref{f8456cb22b9d}. $\blacksquare$
\ No newline at end of file
+\entryref{f8456cb22b9d}. $\blacksquare$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/a4eb0713185a.md b/projects/riemannian-geometry/.astrolabe/atoms/a4eb0713185a.md
index 9a4c16ca..72f13cda 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/a4eb0713185a.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/a4eb0713185a.md
@@ -10,11 +10,11 @@ title: Corollary 2.2
 ---
 Let $M$ and $\tilde{M}$ be spaces with the same constant curvature and the same
 dimension $n$. Let $p\in M$ and $\tilde{p}\in\tilde{M}$. Choose arbitrary
-orthonormal bases $\{e_j\}\in T_p(M)$ and $\{\tilde{e}_j\}\in T_{\tilde{p}}(\tilde{M})$,
+orthonormal bases $\{e_j\}$ of $T_p(M)$ and $\{\tilde{e}_j\}$ of $T_{\tilde{p}}(\tilde{M})$,
 $j=1,\dots,n$. Then there exist a neighborhood $V\subset M$ of $p$, a neighborhood
 $\tilde{V}\subset\tilde{M}$ of $\tilde{p}$, and an isometry $f:V\to\tilde{V}$ such
 that $df_p(e_j)=\tilde{e}_j$.
 
 *Proof.* Choose the isometry $i$ of the theorem in such a way that
 $i(e_j)=\tilde{e}_j$. The condition on the curvature is immediately verified, and
-the conclusion follows from \entryref{4a37954c3621}. $\square$
\ No newline at end of file
+the conclusion follows from \entryref{4a37954c3621}. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/a9965ea4095e.md b/projects/riemannian-geometry/.astrolabe/atoms/a9965ea4095e.md
index 3d1418b8..0f376bf3 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/a9965ea4095e.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/a9965ea4095e.md
@@ -12,8 +12,10 @@ A differentiable manifold $M$ (Hausdorff with countable basis) has a Riemannian
 metric.
 *Proof.* Let $\{f_\alpha\}$ be a differentiable partition of unity subordinate to
 a covering $\{V_\alpha\}$ of $M$ by coordinate neighborhoods (see Chap. 0, Sec. 5),
-so $\{V_\alpha\}$ is locally finite, $f_\alpha\ge0$, $f_\alpha=0$ off
-$\overline{V_\alpha}$, and $\sum_\alpha f_\alpha\equiv1$. On each $V_\alpha$ take
-the metric $\langle\ ,\ \rangle^\alpha$ induced by the local coordinates, and set
-$\langle u,v\rangle_p=\sum_\alpha f_\alpha(p)\langle u,v\rangle_p^\alpha$; this
-defines a Riemannian metric on $M$. $\square$
\ No newline at end of file
+so $\{V_\alpha\}$ is locally finite, $f_\alpha\ge0$,
+$\mathrm{supp}\,f_\alpha\subset V_\alpha$, and
+$\sum_\alpha f_\alpha\equiv1$. On each $V_\alpha$ take the metric
+$\langle\ ,\ \rangle^\alpha$ induced by the local coordinates, and set
+$\langle u,v\rangle_p=\sum_\alpha f_\alpha(p)\langle u,v\rangle_p^\alpha$, where
+only terms with $f_\alpha(p)\ne0$ are evaluated; this defines a Riemannian metric
+on $M$. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/ab0d5657b4ac.md b/projects/riemannian-geometry/.astrolabe/atoms/ab0d5657b4ac.md
index b8e39337..9442e128 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/ab0d5657b4ac.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/ab0d5657b4ac.md
@@ -17,4 +17,6 @@ geodesic through $p$ with velocity $v$; by uniqueness of \entryref{dec3db54a25e}
 follows. Here $\exp_p$ is defined on all of $T_p S^n$: it maps $B_\pi(0)$
 injectively onto $S^n-\{q\}$ ($q$ the antipode of $p$), collapses
 $\partial B_\pi(0)$ to $q$, maps the annulus $B_{2\pi}(0)-\overline{B_\pi(0)}$
-injectively onto $S^n-\{p,q\}$, and collapses $\partial B_{2\pi}(0)$ to $p$.
\ No newline at end of file
+injectively onto $S^n-\{p,q\}$, and collapses $\partial B_{2\pi}(0)$ to $p$.
+For the manifold $S^n-\{q\}$, however, $\exp_p$ is defined only on
+$B_\pi(0)\subset T_p(S^n-\{q\})$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/b7028e5e1b4a.md b/projects/riemannian-geometry/.astrolabe/atoms/b7028e5e1b4a.md
index 4e23496e..058a08c0 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/b7028e5e1b4a.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/b7028e5e1b4a.md
@@ -13,6 +13,11 @@ open, such that $\partial A$ is a piecewise differentiable curve with vertex
 angles different from $\pi$. A *parametrized surface* in $M$ is a differentiable
 map $s:A\subset\mathbb{R}^2\to M$ (differentiable on $A$ meaning extendable to an
 open $U\supset A$; the vertex-angle condition ensures $ds$ is independent of the
-extension). For cartesian $(u,v)$ on $\mathbb{R}^2$, the partial velocity fields
-$\frac{\partial s}{\partial u},\frac{\partial s}{\partial v}$ and covariant
-derivatives $\frac{D}{\partial u},\frac{D}{\partial v}$ are defined along $s$.
\ No newline at end of file
+extension). A vector field $V$ along $s$ assigns to each $q\in A$ a vector
+$V(q)\in T_{s(q)}M$; it is differentiable if $q\mapsto V(q)f$ is differentiable
+for every differentiable function $f$ on $M$. For cartesian $(u,v)$ on
+$\mathbb{R}^2$, $\frac{\partial s}{\partial u}(u_0,v_0)$ is the velocity of
+$u\mapsto s(u,v_0)$ at $u_0$, and $\frac{\partial s}{\partial v}$ is analogous.
+If $V$ is a vector field along $s$, $\frac{DV}{\partial u}(u_0,v_0)$ is the
+covariant derivative along $u\mapsto s(u,v_0)$ of the restriction of $V$, and
+$\frac{DV}{\partial v}$ is defined analogously.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/b75394389199.md b/projects/riemannian-geometry/.astrolabe/atoms/b75394389199.md
index 63b1dfe3..da173c89 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/b75394389199.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/b75394389199.md
@@ -18,10 +18,10 @@ $$
 
 For $r=2$ take $U=B_1(p)$ and $f(q)=\beta(-|p-q|)$, where
 $\beta(t)=\frac{\int_{-\infty}^t\alpha(s)\,ds}{\int_{-2}^{-1}\alpha(s)\,ds}$ and
-$\alpha:\mathbb{R}\to\mathbb{R}$ equals $\exp\big(-\frac{1}{(t+2)(-1-t)}\big)$ on
-$[-2,-1]$ and zero off it. The same holds on $M$: if $V\subset M$ is a
+$\alpha:\mathbb{R}\to\mathbb{R}$ equals $\exp\big(-\frac{1}{(t+2)(-1-t)}\big)$ for
+$-2<t<-1$ and zero otherwise. The same holds on $M$: if $V\subset M$ is a
 neighborhood of $p$ contained in a coordinate neighborhood diffeomorphic to an
 open ball, there exist $U$ with $\overline U\subset V$ and differentiable
 $f:M\to\mathbb{R}$, $0\le f\le1$, $f=1$ on $\overline U$, $f=0$ off $V$. This
 allows showing that certain globally defined objects on $M$ are in reality local
-(cf. the bracket in this chapter and the affine connection in Chapter 2).
\ No newline at end of file
+(cf. the bracket in this chapter and the affine connection in Chapter 2).
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/b9cdcdcd003d.md b/projects/riemannian-geometry/.astrolabe/atoms/b9cdcdcd003d.md
index f23c9ed7..354ed200 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/b9cdcdcd003d.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/b9cdcdcd003d.md
@@ -28,7 +28,13 @@ The spaces of constant curvature have an important role in the historical
 development of Riemannian Geometry, due to their relationship with non-Euclidean
 geometry. A non-Euclidean geometry is a complete Riemannian manifold $M$ together
 with a transitive group of isometries $G$ (the non-Euclidean motions) satisfying
-the *Axiom of free mobility*. This implies that $M$ has constant sectional
-curvature, so the spaces of non-Euclidean Geometry are included among the space
-forms. The cases of $S^n$, $P^n$ and $H^n$ are called *spherical*, *elliptic* and
-*hyperbolic* geometry, respectively.
\ No newline at end of file
+the *Axiom of free mobility*: if $p,\tilde p\in M$, if $\gamma_1,\gamma_2$ are
+geodesics starting at $p$ and forming an angle $\alpha$, and if
+$\tilde\gamma_1,\tilde\gamma_2$ are geodesics starting at $\tilde p$ and forming
+the same angle $\alpha$, then there is a $g\in G$ with $g(p)=\tilde p$,
+$g(\gamma_1)=\tilde\gamma_1$, and $g(\gamma_2)=\tilde\gamma_2$. This corresponds
+to the side-angle-side condition for congruence of triangles in Euclidean geometry
+and implies that $M$ has constant sectional curvature, so the spaces of
+non-Euclidean Geometry are included among the space forms. The cases of $S^n$,
+$P^n$ and $H^n$ are called *spherical*, *elliptic* and *hyperbolic* geometry,
+respectively.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/b9e45fa9e39b.md b/projects/riemannian-geometry/.astrolabe/atoms/b9e45fa9e39b.md
index 62974b44..77f7fb59 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/b9e45fa9e39b.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/b9e45fa9e39b.md
@@ -46,7 +46,7 @@ $T_w(\overset{\circ}{B})$ to a broken geodesic $w$ corresponds to the set of
 Jacobi fields along $w$, broken at $t_i$. It is possible to show that there
 exists a homotopy
 $h_s:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{\Omega}{}^c$, $s\in[0,1]$ with
-$h_o=\mathrm{ident.}$ and $h_1:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{B}$,
+$h_0=\mathrm{ident.}$ and $h_1:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{B}$,
 that is, $\overset{\circ}{B}$ is a "deformation retract" of
 $\overset{\circ}{\Omega}{}^c$.
 
@@ -60,4 +60,4 @@ restricted to the broken Jacobi fields along $\gamma$.
 
 In this way, for many purposes, we can substitute the set
 $\overset{\circ}{\Omega}{}^c$ by its finite dimensional approximation
-$\overset{\circ}{B}$.
\ No newline at end of file
+$\overset{\circ}{B}$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/be5e1aca2bd6.md b/projects/riemannian-geometry/.astrolabe/atoms/be5e1aca2bd6.md
index fadcd9d9..099eca12 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/be5e1aca2bd6.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/be5e1aca2bd6.md
@@ -11,5 +11,5 @@ title: Jacobi
 Let $\gamma:[0,a]\to M$ be a geodesic segment on $M$ such that $\gamma(a)$ is not
 conjugate to $\gamma(0)$. Then $\gamma$ has no conjugate points on $(0,a)$ if and
 only if for all proper variations of $\gamma$ there exists a $\delta>0$ such that
-$E(s)<E(\delta)$ for $0<|s|<\delta$. In particular, if $\gamma$ is minimizing,
-$\gamma$ has no conjugate points on $(0,a)$.
\ No newline at end of file
+$E(0)\leq E(s)$ for $|s|<\delta$. In particular, if $\gamma$ is minimizing,
+$\gamma$ has no conjugate points on $(0,a)$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/c944a1ee39aa.md b/projects/riemannian-geometry/.astrolabe/atoms/c944a1ee39aa.md
index 2d68e19d..55896907 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/c944a1ee39aa.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/c944a1ee39aa.md
@@ -9,9 +9,10 @@ src: docarmo
 title: Proposition 2.4
 ---
 Let $\gamma:[0,a]\to M$ be a geodesic and let $J$ be a Jacobi field along
-$\gamma$ with $J(0)=0$. Put $\frac{DJ}{dt}(0)=w$ and $\gamma'(0)=v$. Consider $w$
-as an element of $T_{av}(T_{\gamma(0)}M)$ and construct a curve $v(s)$ in
-$T_{\gamma(0)}M$ with $v(0)=av$, $v'(0)=w$. Put $f(t,s)=\exp_p(\frac{t}{a}v(s))$,
+$\gamma$ with $J(0)=0$. Put $\frac{DJ}{dt}(0)=w$ and $\gamma'(0)=v$. Using the
+natural identification $T_{av}(T_{\gamma(0)}M)\simeq T_{\gamma(0)}M$, construct a
+curve $v(s)$ in $T_{\gamma(0)}M$ with $v(0)=av$, $v'(0)=aw$. Put
+$f(t,s)=\exp_p(\frac{t}{a}v(s))$,
 $p=\gamma(0)$, and define a Jacobi field $\bar J$ by
 $\bar J(t)=\frac{\partial f}{\partial s}(t,0)$. Then $\bar J=J$ on $[0,a]$.
 
@@ -30,4 +31,4 @@ $$
 $$
 
 Since $J(0)=\bar J(0)=0$ and $\frac{DJ}{dt}(0)=\frac{D\bar J}{dt}(0)=w$, we
-conclude, from the uniqueness theorem, that $J=\bar J$. $\square$
\ No newline at end of file
+conclude, from the uniqueness theorem, that $J=\bar J$. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/cb5eff54c665.md b/projects/riemannian-geometry/.astrolabe/atoms/cb5eff54c665.md
index e9831295..5163f1ba 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/cb5eff54c665.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/cb5eff54c665.md
@@ -24,7 +24,7 @@ $$
 for all $X,Y\in\mathcal{X}(M)$, $f,g\in\mathcal{D}(M)$.
 
 A tensor is a pointwise object. Fix $p\in M$ and let $U$ be a neighborhood of $p$
-on which it is possible to define vector fields $E_1,\dots,E_n\in\mathcal{X}(M^n)$
+on which it is possible to define vector fields $E_1,\dots,E_n\in\mathcal{X}(U)$
 such that at each $q\in U$ the vectors $\{E_i(q)\}$, $i=1,\dots,n$, form a basis of
 $T_qM$; we say $\{E_i\}$ is a *moving frame* on $U$. Writing
 $Y_1=\sum_{i_1}y_{i_1}E_{i_1},\dots,Y_r=\sum_{i_r}y_{i_r}E_{i_r}$, by linearity
@@ -36,4 +36,4 @@ $$
 The functions $T(E_{i_1},\dots,E_{i_r})=T_{i_1\dots i_r}$ on $U$ are called the
 *components* of $T$ in the frame $\{E_i\}$. The value of $T(Y_1,\dots,Y_r)$ at $p$
 depends only on the values at $p$ of the components of $T$ and the values of
-$Y_1,\dots,Y_r$ at $p$.
\ No newline at end of file
+$Y_1,\dots,Y_r$ at $p$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/d483a90d3a7d.md b/projects/riemannian-geometry/.astrolabe/atoms/d483a90d3a7d.md
index 412005a5..4fe57f0c 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/d483a90d3a7d.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/d483a90d3a7d.md
@@ -42,6 +42,8 @@ $$
 \tilde J(t)=\frac{\sinh(t\sqrt{-b})}{\sinh(\ell\sqrt{-b})}\tilde w(t)\ (b<0),\qquad\tilde J(t)=\frac{t}{\ell}\tilde w(t)\ (b=0),
 $$
 
+where $\tilde w$ is the parallel vector field along $\tilde\gamma$ with
+$\tilde w(\ell)=\tilde v$. Thus
 one obtains from the proof of the Index Lemma
 
 $$
@@ -75,4 +77,4 @@ $$
 $$
 
 That (6) and (7) are incompatible with $\dim\overline{M}<2\dim M$ follows from
-\entryref{e42a427100fe}, completing the proof. $\blacksquare$
\ No newline at end of file
+\entryref{e42a427100fe}, completing the proof. $\blacksquare$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/db5c231a456e.md b/projects/riemannian-geometry/.astrolabe/atoms/db5c231a456e.md
index 745f18cc..de23f977 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/db5c231a456e.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/db5c231a456e.md
@@ -15,7 +15,8 @@ every proper variation $f$ of $c$, we have $\frac{dE}{ds}(0)=0$.
 for $f$ proper, $V(0)=V(a)=0$, and all terms of (1) are zero. Conversely, suppose
 $\frac{dE}{ds}(0)=0$ for all proper variations. Let
 $V(t)=g(t)\frac{D}{dt}\frac{dc}{dt}$, where $g:[0,a]\to\mathbf{R}$ is piecewise
-differentiable with $g(t)>0$ if $t\ne t_i$ and $g(t_i)=0$. Constructing a variation
+differentiable with $g(t)>0$ if $t\ne t_i$ and $g(t_i)=0$, $i=0,\dots,k+1$.
+Constructing a variation
 with this $V(t)$,
 
 $$
@@ -33,4 +34,4 @@ $$
 
 so $c$ is of class $C^1$ at each $t_i$. Since $\frac{D}{dt}\frac{dc}{dt}=0$ at
 $t_i$, $c$ satisfies the geodesic equation on $(0,a)$; by uniqueness of solutions
-to ordinary differential equations, $c\in C^\infty$ and is a geodesic. $\square$
\ No newline at end of file
+to ordinary differential equations, $c\in C^\infty$ and is a geodesic. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/e51da6670107.md b/projects/riemannian-geometry/.astrolabe/atoms/e51da6670107.md
index 55744d9d..2b4ee0b7 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/e51da6670107.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/e51da6670107.md
@@ -12,7 +12,9 @@ A *Riemannian metric* (or *Riemannian structure*) on a differentiable manifold
 $M$ associates to each $p\in M$ an inner product $\langle\ ,\ \rangle_p$ (a
 symmetric, bilinear, positive-definite form) on $T_pM$, varying differentiably:
 if $\mathbf{x}:U\subset\mathbb{R}^n\to M$ is a coordinate system around $p$ with
-$\frac{\partial}{\partial x_i}(q)=d\mathbf{x}_q(0,\dots,1,\dots,0)$, then
+$q=\mathbf{x}(x_1,\dots,x_n)\in\mathbf{x}(U)$ and
+$\frac{\partial}{\partial x_i}(q)=d\mathbf{x}_{(x_1,\dots,x_n)}(0,\dots,1,\dots,0)$,
+then
 
 $$
 g_{ij}(x_1,\dots,x_n)=\Big\langle\frac{\partial}{\partial x_i}(q),\frac{\partial}{\partial x_j}(q)\Big\rangle_q
@@ -22,4 +24,4 @@ is a differentiable function on $U$. The definition is independent of the chart;
 equivalently, $\langle X,Y\rangle$ is differentiable for any differentiable
 vector fields $X,Y$ on $V\subset M$. The functions $g_{ij}=g_{ji}$ are the *local
 representation of the metric*. A differentiable manifold with a given Riemannian
-metric is a *Riemannian manifold*.
\ No newline at end of file
+metric is a *Riemannian manifold*.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/e5ec3204f4f6.md b/projects/riemannian-geometry/.astrolabe/atoms/e5ec3204f4f6.md
index bca22d6d..9f14a976 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/e5ec3204f4f6.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/e5ec3204f4f6.md
@@ -18,7 +18,7 @@ dimension $n-m=k$, hence (\entryref{2a37e4843dff}) a submanifold of $\mathbb{R}^
 *Proof.* For $p\in F^{-1}(a)$, writing $q=(y_1,\dots,y_m,x_1,\dots,x_k)$ and
 $F(q)=(f_1(q),\dots,f_m(q))$, since $dF_p$ is surjective suppose
 $\frac{\partial(f_1,\dots,f_m)}{\partial(y_1,\dots,y_m)}(p)\neq0$. Define
-$\varphi:U\to\mathbb{R}^{n=m+k}$ by
+$\varphi:U\to\mathbb{R}^{m+k}$ by
 $\varphi(y,x)=(f_1(q),\dots,f_m(q),x_1,\dots,x_k)$, with
 $\det(d\varphi)_p=\frac{\partial(f_1,\dots,f_m)}{\partial(y_1,\dots,y_m)}(p)\neq0$.
 By the inverse function theorem $\varphi$ is a diffeomorphism of a neighborhood
@@ -26,4 +26,4 @@ $Q$ of $p$ onto $W$; with a cube $K^{m+k}$ centered at $\varphi(p)$ and
 $V=\varphi^{-1}(K^{m+k})\cap Q$, $\varphi$ maps $V$ diffeomorphically onto
 $K^m\times K^k$, and $\mathbf{x}:K^k\to V$,
 $\mathbf{x}(x_1,\dots,x_k)=\varphi^{-1}(a_1,\dots,a_m,x_1,\dots,x_k)$, satisfies
-(a),(b) of \entryref{2a37e4843dff}. Hence $F^{-1}(a)$ is a regular surface. $\square$
\ No newline at end of file
+(a),(b) of \entryref{2a37e4843dff}. Hence $F^{-1}(a)$ is a regular surface. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/e8d487d40068.md b/projects/riemannian-geometry/.astrolabe/atoms/e8d487d40068.md
index 7dc82306..e10c2ece 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/e8d487d40068.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/e8d487d40068.md
@@ -17,7 +17,9 @@ X(p)=\sum_{i=1}^n a_i(p)\frac{\partial}{\partial x_i},\qquad\text{(4)}
 $$
 
 $a_i:U\to\mathbb{R}$; $X$ is differentiable iff the $a_i$ are differentiable for
-some (hence any) parametrization. Thinking of $X:\mathcal{D}\to\mathcal{F}$,
+some (hence any) parametrization. Let $\mathcal{D}$ be the differentiable
+functions on $M$ and $\mathcal{F}$ the functions on $M$; thinking of
+$X:\mathcal{D}\to\mathcal{F}$,
 
 $$
 (Xf)(p)=\sum_i a_i(p)\frac{\partial f}{\partial x_i}(p),\qquad\text{(5)}
@@ -26,4 +28,4 @@ $$
 $Xf$ is independent of $\mathbf{x}$, and $X$ is differentiable iff
 $Xf\in\mathcal{D}$ for all $f\in\mathcal{D}$. If $\varphi:M\to M$ is a
 diffeomorphism, $v\in T_pM$, and $f$ differentiable near $\varphi(p)$, then
-$(d\varphi(v)f)\varphi(p)=v(f\circ\varphi)(p)$.
\ No newline at end of file
+$(d\varphi(v)f)\varphi(p)=v(f\circ\varphi)(p)$.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/ed5c99ea7483.md b/projects/riemannian-geometry/.astrolabe/atoms/ed5c99ea7483.md
index bd85c3e3..c1bd3513 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/ed5c99ea7483.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/ed5c99ea7483.md
@@ -66,13 +66,17 @@ transformations.
 If $M$ is a Riemannian manifold and $\Gamma$ is a subgroup of the group of
 isometries acting in a totally discontinuous manner, then $M/\Gamma$ has a
 differentiable structure in which $\pi:M\to M/\Gamma$ is a local diffeomorphism,
-and a Riemannian metric making $\pi$ a local isometry, defined by
+and a Riemannian metric making $\pi$ a local isometry. Given $p\in M/\Gamma$,
+choose $\tilde p\in\pi^{-1}(p)$; for $u,v\in T_p(M/\Gamma)$, define
 
 $$
 \langle u,v\rangle=\langle d\pi^{-1}(u),d\pi^{-1}(v)\rangle_{\tilde{p}},
 $$
 
-called the *metric on $M/\Gamma$ induced by the covering $\pi$*. $M/\Gamma$ is
-complete iff $M$ is, and has constant curvature iff $M$ does. Taking $M=S^n$,
+Since $\pi$ is a regular covering, $\Gamma$ is transitive on $\pi^{-1}(p)$, and
+because $\Gamma$ consists of isometries the definition is independent of the
+choice of $\tilde p$. This is called the *metric on $M/\Gamma$ induced by the
+covering $\pi$*. $M/\Gamma$ is complete iff $M$ is, and has constant curvature iff
+$M$ does. Taking $M=S^n$,
 $\mathbb{R}^n$ or $H^n$, we conclude that $M/\Gamma$ is a complete manifold of
-constant curvature $1$, $0$ or $-1$, respectively.
\ No newline at end of file
+constant curvature $1$, $0$ or $-1$, respectively.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/ef35a77f11bf.md b/projects/riemannian-geometry/.astrolabe/atoms/ef35a77f11bf.md
index 1a8b6fc6..9a74022b 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/ef35a77f11bf.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/ef35a77f11bf.md
@@ -18,9 +18,10 @@ since $\langle df_p(v),df_p(v)\rangle=16\langle v,v\rangle/|p-p_0|^4$ and
 $f_n(p)=(4-|p|^2)/|p-p_0|^2$ give
 $\langle df_p(v),df_p(v)\rangle/(f_n(p))^2=\langle v,v\rangle/(1-\tfrac14|p|^2)^2$.
 Because $f$ is injective, $f$ is an isometry of $B^n$ into $H^n$, taking
-$\partial B^n-\{p_0\}$ into $\partial H^n$. By Liouville's Theorem, $g:B^n\to B^n$
-is an isometry in $h_{ij}$ iff $g$ is the restriction of a conformal transformation
-of $\mathbb{R}^n$ taking $B^n$ onto $B^n$.
+$\partial B^n-\{p_0\}$ into $\partial H^n$. Conjugating by (13) and applying
+\entryref{8a57517fee19}, $g:B^n\to B^n$ is an isometry in $h_{ij}$ iff $g$ is the
+restriction of a conformal transformation of $\mathbb{R}^n$ taking $B^n$ onto
+$B^n$.
 
 Let $S\subset H^n$ be an $(n-1)$-sphere of Euclidean space completely contained in
 $H^n$. Then $S$ is a *geodesic sphere* of $H^n$: its image $f^{-1}(S)\subset B^n$ is
@@ -35,15 +36,17 @@ metric on $P$ is a multiple of the Euclidean metric, so $P$ has constant curvatu
 zero, as does $S-\{p\}$. Such submanifolds are called *horospheres* of $H^n$.
 
 Let $S$ be a Euclidean sphere cutting $\partial H^n$ at an angle $\alpha$, and
-denote $S\cap H^n=\sum$. Through an inversion at a point of $S\cap\partial H^n$,
-$\sum$ is mapped isometrically into the intersection with $H^n$ of a hyperplane $P$
+denote $S\cap H^n=\Sigma$. Through an inversion at a point of $S\cap\partial H^n$,
+$\Sigma$ is mapped isometrically into the intersection with $H^n$ of a hyperplane $P$
 cutting $\partial H^n$ at the same angle $\alpha$. Let $Q$ be the hyperplane
 orthogonal to $\partial H^n$ containing $P\cap\partial H^n$. Then $P$ is a
 hypersurface equidistant from the totally geodesic hypersurface $Q$: for a geodesic
-$\gamma_r$ (semi-circle of radius $r$ with center $0\in P\cap\partial H^n$), a
+$\gamma_r$ (a semi-circle of radius $r$ with center $0\in P\cap\partial H^n$,
+lying in the plane perpendicular to $P\cap\partial H^n$), a
 homothety with center $0$ takes the circle of radius $r$ into one of any radius, so
 the length of $\gamma_r$ between its intersections with $P$ and $Q$ does not depend
-on $r$. We conclude that $P$, or its isometric image $\sum$, is obtained by taking
+on $r$. We conclude that $P$, or its isometric image $\Sigma$, is obtained by taking
 geodesics perpendicular to $Q$ and marking a fixed distance. Such hypersurfaces are
 called *equidistant surfaces* (or *hyperspheres*). In Exercise 6 it is shown that
-the hypersurfaces above are exactly the umbilic ones.
\ No newline at end of file
+the hypersurfaces above are exactly the umbilic ones, and their mean and sectional
+curvatures are calculated.
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/fa0c59ada60c.md b/projects/riemannian-geometry/.astrolabe/atoms/fa0c59ada60c.md
index 60c72494..dfd4871f 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/fa0c59ada60c.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/fa0c59ada60c.md
@@ -19,8 +19,10 @@ $$
 $d\exp_p$ is linear and
 $\langle(d\exp_p)_v(v),(d\exp_p)_v(w_T)\rangle=\langle v,w_T\rangle$, it suffices
 to prove (2) for $w=w_N$ ($w_N\ne 0$). Pick a curve $v(s)$ in $T_pM$ with
-$v(0)=v$, $v'(0)=w_N$, $|v(s)|=\text{const}$, and consider the parametrized
-surface $f(t,s)=\exp_p tv(s)$ whose curves $t\mapsto f(t,s_0)$ are geodesics.
+$v(0)=v$, $v'(0)=w_N$, $|v(s)|=\text{const}$. Since $\exp_p v$ is defined,
+choose $\varepsilon>0$ so that $\exp_p(tv(s))$ is defined for $0\le t\le 1$,
+$|s|<\varepsilon$, and consider the parametrized surface
+$f(t,s)=\exp_p tv(s)$ whose curves $t\mapsto f(t,s_0)$ are geodesics.
 Then $\langle\frac{\partial f}{\partial s},\frac{\partial f}{\partial t}\rangle(1,0)=\langle(d\exp_p)_v(w_N),(d\exp_p)_v(v)\rangle$.
 Since $\frac{\partial f}{\partial t}$ is the tangent of a geodesic and by symmetry
 of the connection,
@@ -31,4 +33,4 @@ $$
 
 so $\langle\frac{\partial f}{\partial s},\frac{\partial f}{\partial t}\rangle$ is
 independent of $t$. As $\lim_{t\to0}\frac{\partial f}{\partial s}(t,0)=\lim_{t\to0}(d\exp_p)_{tv}\,t w_N=0$,
-the inner product is $0$, proving (2). $\square$
\ No newline at end of file
+the inner product is $0$, proving (2). $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/atoms/ff0d95de92e4.md b/projects/riemannian-geometry/.astrolabe/atoms/ff0d95de92e4.md
index 292c987e..e17b3736 100644
--- a/projects/riemannian-geometry/.astrolabe/atoms/ff0d95de92e4.md
+++ b/projects/riemannian-geometry/.astrolabe/atoms/ff0d95de92e4.md
@@ -28,12 +28,13 @@ $\bar\gamma_1$ off $L$ within $H$ fixing endpoints; the gradient flow then retra
 into $M^{c_1-\varepsilon}$. Proceeding inductively yields
 $\bar\gamma\subset M^{a+\delta}$. $\square$
 
-(Remark after Lemma 3.3: if instead the largest critical value $c$ in $f^{-1}([a,b])$
-has index zero or one, then $\gamma$ is homotopic to $\bar\gamma$ with
+(Remark after Lemma 3.3: if there are critical points of index zero or one in
+$f^{-1}([a,b])$ and $c$ is the largest value of such critical points, then
+$\gamma$ is homotopic to $\bar\gamma$ with
 $\bar\gamma([0,1])\subset M^{c+\delta}$.)
 
 *Proof of \entryref{615bcaf45bdc}.* Suppose $i(M)<\pi$. By \entryref{2660f4a30491} there is a
-closed geodesic $\gamma$ of length $\ell=\ell(\gamma)<2\pi$. By \entryref{9b500d558b2c} of
+closed geodesic $\gamma$ of length $\ell=\ell(\gamma)<2\pi$. By \entryref{2b808289bf64} of
 Chap. 11 the conjugate points to $\gamma(0)$ along $\gamma$ are discrete; choose
 $\varepsilon>0$ with: (1) $\gamma(\ell-\varepsilon)$ not conjugate to $p=\gamma(0)$;
 (2) $\exp_p$ a diffeomorphism on $B_{2\varepsilon}(p)$;
@@ -45,7 +46,7 @@ minimizing geodesic $\gamma_o$ from $p$ to $q$ has
 $\ell(\gamma_o)\leq d(p,\gamma(\ell-\varepsilon))+d(\gamma(\ell-\varepsilon),q)\leq 2\varepsilon$,
 so $\gamma_o\neq\gamma_1$.
 
-Consider $\Omega_{p,q}^c$ (\entryref{d8300a62544d} of Chap. 11) and its finite-dimensional
+Consider $\Omega_{p,q}^c$ (\entryref{b9e45fa9e39b} of Chap. 11) and its finite-dimensional
 approximation $B$. Since $q$ is a regular value, all critical points in
 $\Omega_{p,q}^c$ are non-degenerate. As $M$ is simply connected, there is a homotopy
 $\gamma_s$ between $\gamma_o$ and $\gamma_1$ fixing $p,q$; deforming into $B$ and
@@ -63,4 +64,4 @@ $\ell(\bar\gamma)<a+\varepsilon<2\pi-3\varepsilon+\varepsilon=2\pi-2\varepsilon$
 the other hand, the Lemma of Klingenberg (Exercise 1 of Chap. 10) gives in the
 homotopy a curve $\bar\gamma_{s_o}$ with $\ell(\gamma_o)+\ell(\bar\gamma_{s_o})\geq2\pi$,
 so $\ell(\bar\gamma)\geq2\pi-\ell(\gamma_o)\geq2\pi-2\varepsilon$, contradicting
-$\ell(\bar\gamma)<2\pi-2\varepsilon$. Thus $i(M)<\pi$ is untenable. $\square$
\ No newline at end of file
+$\ell(\bar\gamma)<2\pi-2\varepsilon$. Thus $i(M)<\pi$ is untenable. $\square$
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/00-manifolds.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/00-manifolds.mdx
index 724a5ad4..186ac3d5 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/00-manifolds.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/00-manifolds.mdx
@@ -295,7 +295,7 @@ dimension $n-m=k$, hence (Example 4.2) a submanifold of $\mathbb{R}^n$.
 *Proof.* For $p\in F^{-1}(a)$, writing $q=(y_1,\dots,y_m,x_1,\dots,x_k)$ and
 $F(q)=(f_1(q),\dots,f_m(q))$, since $dF_p$ is surjective suppose
 $\frac{\partial(f_1,\dots,f_m)}{\partial(y_1,\dots,y_m)}(p)\neq0$. Define
-$\varphi:U\to\mathbb{R}^{n=m+k}$ by
+$\varphi:U\to\mathbb{R}^{m+k}$ by
 $\varphi(y,x)=(f_1(q),\dots,f_m(q),x_1,\dots,x_k)$, with
 $\det(d\varphi)_p=\frac{\partial(f_1,\dots,f_m)}{\partial(y_1,\dots,y_m)}(p)\neq0$.
 By the inverse function theorem $\varphi$ is a diffeomorphism of a neighborhood
@@ -336,9 +336,12 @@ $\pi_1(x)=\big(\frac{x_1}{1-x_{n+1}},\dots,\frac{x_n}{1-x_{n+1}}\big)$, and
 $\pi_2:S^n-\{S\}\to\mathbb{R}^n$ (from the south pole) are differentiable
 injections onto $x_{n+1}=0$. The parametrizations
 $(\mathbb{R}^n,\pi_1^{-1}),(\mathbb{R}^n,\pi_2^{-1})$ cover $S^n$ with change of
-coordinates $y_j'=\frac{y_j}{\sum_{i=1}^n y_i^2}$, differentiable, and
-$\pi_1^{-1}(\mathbb{R}^n)\cap\pi_2^{-1}(\mathbb{R}^n)=S^n-\{N\cup S\}$ connected;
-by Example 4.5, $S^n$ is orientable. The antipodal map $A:S^n\to S^n$, $A(p)=-p$,
+coordinates $y_j'=\frac{y_j}{\sum_{i=1}^n y_i^2}$, differentiable. For $n\geq2$,
+$\pi_1^{-1}(\mathbb{R}^n)\cap\pi_2^{-1}(\mathbb{R}^n)=S^n-\{N,S\}$ is connected,
+so by Example 4.5, $S^n$ is orientable. For $n=1$, the same atlas is orientable
+directly: on the two components of $S^1-\{N,S\}$, the change of coordinates is
+$y'=1/y$ with derivative $-1/y^2<0$, and changing the sign of one coordinate makes
+both transition derivatives positive. The antipodal map $A:S^n\to S^n$, $A(p)=-p$,
 is a diffeomorphism ($A^2=\mathrm{id}$); it reverses the orientation of $S^n$ when
 $n$ is even and preserves it when $n$ is odd.
 
@@ -385,8 +388,9 @@ situation of Example 4.7 is this one with $M=S^n$, $G=\{A,I=A^2\}$.
 
 ### 4.9 Example (special cases of Example 4.8)
 
-**4.9 (a) (the $k$-torus).** $G$ = group of "integral" translations of
-$\mathbb{R}^k$, $G(x_1,\dots,x_k)=(x_1+n_1,\dots,x_k+n_k)$, $n_i\in\mathbb{Z}$.
+**4.9 (a) (the $k$-torus).** Let $G$ be the group of "integral" translations of
+$\mathbb{R}^k$: for $(n_1,\dots,n_k)\in\mathbb{Z}^k$, the corresponding
+translation sends $(x_1,\dots,x_k)$ to $(x_1+n_1,\dots,x_k+n_k)$.
 This is a properly discontinuous action; $\mathbb{R}^k/G$ (with the structure of
 Example 4.8) is the *$k$-torus $T^k$*. For $k=2$, $T^2$ is diffeomorphic to the
 torus of revolution in $\mathbb{R}^3$, the inverse image of $0$ of
@@ -412,7 +416,9 @@ X(p)=\sum_{i=1}^n a_i(p)\frac{\partial}{\partial x_i},\qquad\text{(4)}
 $$
 
 $a_i:U\to\mathbb{R}$; $X$ is differentiable iff the $a_i$ are differentiable for
-some (hence any) parametrization. Thinking of $X:\mathcal{D}\to\mathcal{F}$,
+some (hence any) parametrization. Let $\mathcal{D}$ be the differentiable
+functions on $M$ and $\mathcal{F}$ the functions on $M$; thinking of
+$X:\mathcal{D}\to\mathcal{F}$,
 
 $$
 (Xf)(p)=\sum_i a_i(p)\frac{\partial f}{\partial x_i}(p),\qquad\text{(5)}
@@ -542,8 +548,8 @@ $$
 
 For $r=2$ take $U=B_1(p)$ and $f(q)=\beta(-|p-q|)$, where
 $\beta(t)=\frac{\int_{-\infty}^t\alpha(s)\,ds}{\int_{-2}^{-1}\alpha(s)\,ds}$ and
-$\alpha:\mathbb{R}\to\mathbb{R}$ equals $\exp\big(-\frac{1}{(t+2)(-1-t)}\big)$ on
-$[-2,-1]$ and zero off it. The same holds on $M$: if $V\subset M$ is a
+$\alpha:\mathbb{R}\to\mathbb{R}$ equals $\exp\big(-\frac{1}{(t+2)(-1-t)}\big)$ for
+$-2<t<-1$ and zero otherwise. The same holds on $M$: if $V\subset M$ is a
 neighborhood of $p$ contained in a coordinate neighborhood diffeomorphic to an
 open ball, there exist $U$ with $\overline U\subset V$ and differentiable
 $f:M\to\mathbb{R}$, $0\le f\le1$, $f=1$ on $\overline U$, $f=0$ off $V$. This
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/01-metrics.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/01-metrics.mdx
index a54fabc2..2b5618c6 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/01-metrics.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/01-metrics.mdx
@@ -29,7 +29,9 @@ A *Riemannian metric* (or *Riemannian structure*) on a differentiable manifold
 $M$ associates to each $p\in M$ an inner product $\langle\ ,\ \rangle_p$ (a
 symmetric, bilinear, positive-definite form) on $T_pM$, varying differentiably:
 if $\mathbf{x}:U\subset\mathbb{R}^n\to M$ is a coordinate system around $p$ with
-$\frac{\partial}{\partial x_i}(q)=d\mathbf{x}_q(0,\dots,1,\dots,0)$, then
+$q=\mathbf{x}(x_1,\dots,x_n)\in\mathbf{x}(U)$ and
+$\frac{\partial}{\partial x_i}(q)=d\mathbf{x}_{(x_1,\dots,x_n)}(0,\dots,1,\dots,0)$,
+then
 
 $$
 g_{ij}(x_1,\dots,x_n)=\Big\langle\frac{\partial}{\partial x_i}(q),\frac{\partial}{\partial x_j}(q)\Big\rangle_q
@@ -111,12 +113,14 @@ differential $\mathrm{Ad}(a)=dR_{a^{-1}}dL_a:\mathcal{G}\to\mathcal{G}$, so
 $\mathrm{Ad}(a)Y=dR_{a^{-1}}Y$. Let $x_t$ be the flow of $X$. From Proposition 5.4
 of Chapter 0, $[Y,X]=\lim_{t\to0}\frac1t(dx_t(Y)-Y)$. Since $X$ is left invariant,
 $L_y\circ x_t=x_t\circ L_y$, giving $x_t(y)=R_{x_t(e)}(y)$, hence $dx_t=dR_{x_t(e)}$
-and $[Y,X]=\lim_{t\to0}\frac1t(\mathrm{Ad}(x_t^{-1}(e))Y-Y)$. With $\langle\ ,\ \rangle$
+and $[Y,X]=\lim_{t\to0}\frac1t(\mathrm{Ad}((x_t(e))^{-1})Y-Y)$. With $\langle\ ,\ \rangle$
 bi-invariant, $\langle U,V\rangle=\langle dR_{x_t(e)}U,dR_{x_t(e)}V\rangle$;
 differentiating in $t$ at $t=0$ gives $0=\langle[U,X],V\rangle+\langle U,[V,X]\rangle$,
-which is (3). The relation (3) characterizes bi-invariant metrics: a positive
-bilinear form on $\mathcal{G}$ satisfying (3) yields, via (2), a bi-invariant
-metric. $\blacksquare$
+which is (3). $\blacksquare$
+
+The relation (3) characterizes bi-invariant metrics: a positive bilinear form on
+$\mathcal{G}$ satisfying (3) yields, via (2), a bi-invariant metric. This last
+fact is not proved here.
 
 ### 2.7 Example (the product metric)
 Let $M_1,M_2$ be Riemannian manifolds and $\pi_1,\pi_2$ the natural projections
@@ -152,25 +156,13 @@ A differentiable manifold $M$ (Hausdorff with countable basis) has a Riemannian
 metric.
 *Proof.* Let $\{f_\alpha\}$ be a differentiable partition of unity subordinate to
 a covering $\{V_\alpha\}$ of $M$ by coordinate neighborhoods (see Chap. 0, Sec. 5),
-so $\{V_\alpha\}$ is locally finite, $f_\alpha\ge0$, $f_\alpha=0$ off
-$\overline{V_\alpha}$, and $\sum_\alpha f_\alpha\equiv1$. On each $V_\alpha$ take
-the metric $\langle\ ,\ \rangle^\alpha$ induced by the local coordinates, and set
-$\langle u,v\rangle_p=\sum_\alpha f_\alpha(p)\langle u,v\rangle_p^\alpha$; this
-defines a Riemannian metric on $M$. $\square$
-
-### 2.11 Remark (volume form)
-The reader familiar with differential forms will note that equation (4) implies
-the integrand in the volume formula (5) is a positive differential form of degree
-$n$, the *volume form (volume element)* $\nu$ on $M$. For a compact region $R$ not
-contained in one coordinate neighborhood, take a partition of unity $\{\varphi_i\}$
-subordinate to a finite covering by coordinate neighborhoods $\mathbf{x}(U_i)$ and
-set $\mathrm{vol}(R)=\sum_i\int_{\mathbf{x}_i^{-1}(R)}\varphi_i\nu$, independent of
-the partition.
-
-### 2.12 Remark (volume from a volume element)
-The existence of a globally defined positive differential form of degree $n$
-(volume element) yields a notion of volume on a differentiable manifold. A
-Riemannian metric is only one of the ways a volume element can be obtained.
+so $\{V_\alpha\}$ is locally finite, $f_\alpha\ge0$,
+$\mathrm{supp}\,f_\alpha\subset V_\alpha$, and
+$\sum_\alpha f_\alpha\equiv1$. On each $V_\alpha$ take the metric
+$\langle\ ,\ \rangle^\alpha$ induced by the local coordinates, and set
+$\langle u,v\rangle_p=\sum_\alpha f_\alpha(p)\langle u,v\rangle_p^\alpha$, where
+only terms with $f_\alpha(p)\ne0$ are evaluated; this defines a Riemannian metric
+on $M$. $\square$
 
 ### Volume on an oriented manifold (between 2.10 and 2.11)
 For a positive parametrization $\mathbf{x}:U\to M$ about $p$, take a positive
@@ -186,7 +178,7 @@ $$
 For another positive parametrization $\mathbf{y}:V\to M$ with $h_{ij}=\langle Y_i,Y_j\rangle$,
 
 $$
-\sqrt{\det(g_{ij})}(p)=J\sqrt{\det(h_{ij})}(p),\qquad J=\det\big(\tfrac{\partial x_i}{\partial y_j}\big)>0.\qquad\text{(4)}
+\sqrt{\det(g_{ij})}(p)=J\sqrt{\det(h_{ij})}(p),\qquad J=\det\big(\tfrac{\partial y_i}{\partial x_j}\big)>0.\qquad\text{(4)}
 $$
 
 For a region $R\subset M$ (open, connected, compact closure) contained in a
@@ -199,3 +191,17 @@ $$
 
 By the change-of-variables theorem and (4) this is independent of the chart
 (orientability guarantees $\mathrm{vol}(R)$ does not change sign).
+
+### 2.11 Remark (volume form)
+The reader familiar with differential forms will note that equation (4) implies
+the integrand in the volume formula (5) is a positive differential form of degree
+$n$, the *volume form (volume element)* $\nu$ on $M$. For a compact region $R$ not
+contained in one coordinate neighborhood, take a partition of unity $\{\varphi_i\}$
+subordinate to a finite covering by coordinate neighborhoods $\mathbf{x}(U_i)$ and
+set $\mathrm{vol}(R)=\sum_i\int_{\mathbf{x}_i^{-1}(R)}\varphi_i\nu$, independent of
+the partition.
+
+### 2.12 Remark (volume from a volume element)
+The existence of a globally defined positive differential form of degree $n$
+(volume element) yields a notion of volume on a differentiable manifold. A
+Riemannian metric is only one of the ways a volume element can be obtained.
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/02-connections.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/02-connections.mdx
index 4a633c92..026732dc 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/02-connections.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/02-connections.mdx
@@ -101,10 +101,12 @@ $c(t_0)$, $t_0\in I$ (i.e. $V_0\in T_{c(t_0)}M$). Then there exists a unique
 parallel vector field $V$ along $c$ with $V(t_0)=V_0$; $V(t)$ is called the
 *parallel transport* of $V(t_0)$ along $c$.
 
-*Proof.* By compactness $c([t_0,t_1])$ can be covered by finitely many coordinate
-neighborhoods, and uniqueness makes the local definitions coincide on overlaps,
-so it suffices to prove the theorem when $c(I)$ lies in one coordinate
-neighborhood $\mathbf{x}(U)$. Writing $V=\sum v^j X_j$, parallelism gives
+*Proof.* Suppose first that the theorem is proved when $c(I)$ is contained in a
+local coordinate neighborhood. By compactness, for any $t_1\in I$,
+$c([t_0,t_1])$ can be covered by finitely many coordinate neighborhoods, and
+uniqueness makes the local definitions coincide on overlaps, so it suffices to
+prove the theorem when $c(I)$ lies in one coordinate neighborhood
+$\mathbf{x}(U)$. Writing $V=\sum v^j X_j$, parallelism gives
 $0=\frac{DV}{dt}=\sum_j\frac{dv^j}{dt}X_j+\sum_{i,j}\frac{dx_i}{dt}v^j\nabla_{X_i}X_j$.
 Putting $\nabla_{X_i}X_j=\sum_k\Gamma_{ij}^k X_k$ yields the system of $n$ linear
 ODEs in $v^k(t)$,
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/03-geodesics.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/03-geodesics.mdx
index 3e61a952..c40ab9d5 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/03-geodesics.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/03-geodesics.mdx
@@ -218,6 +218,8 @@ follows. Here $\exp_p$ is defined on all of $T_p S^n$: it maps $B_\pi(0)$
 injectively onto $S^n-\{q\}$ ($q$ the antipode of $p$), collapses
 $\partial B_\pi(0)$ to $q$, maps the annulus $B_{2\pi}(0)-\overline{B_\pi(0)}$
 injectively onto $S^n-\{p,q\}$, and collapses $\partial B_{2\pi}(0)$ to $p$.
+For the manifold $S^n-\{q\}$, however, $\exp_p$ is defined only on
+$B_\pi(0)\subset T_p(S^n-\{q\})$.
 
 ## §3 Minimizing properties of geodesics
 
@@ -240,9 +242,14 @@ open, such that $\partial A$ is a piecewise differentiable curve with vertex
 angles different from $\pi$. A *parametrized surface* in $M$ is a differentiable
 map $s:A\subset\mathbb{R}^2\to M$ (differentiable on $A$ meaning extendable to an
 open $U\supset A$; the vertex-angle condition ensures $ds$ is independent of the
-extension). For cartesian $(u,v)$ on $\mathbb{R}^2$, the partial velocity fields
-$\frac{\partial s}{\partial u},\frac{\partial s}{\partial v}$ and covariant
-derivatives $\frac{D}{\partial u},\frac{D}{\partial v}$ are defined along $s$.
+extension). A vector field $V$ along $s$ assigns to each $q\in A$ a vector
+$V(q)\in T_{s(q)}M$; it is differentiable if $q\mapsto V(q)f$ is differentiable
+for every differentiable function $f$ on $M$. For cartesian $(u,v)$ on
+$\mathbb{R}^2$, $\frac{\partial s}{\partial u}(u_0,v_0)$ is the velocity of
+$u\mapsto s(u,v_0)$ at $u_0$, and $\frac{\partial s}{\partial v}$ is analogous.
+If $V$ is a vector field along $s$, $\frac{DV}{\partial u}(u_0,v_0)$ is the
+covariant derivative along $u\mapsto s(u,v_0)$ of the restriction of $V$, and
+$\frac{DV}{\partial v}$ is defined analogously.
 
 ### 3.4 Lemma (symmetry)
 If $M$ has a symmetric connection and $s:A\to M$ is a parametrized surface, then
@@ -277,8 +284,10 @@ $$
 $d\exp_p$ is linear and
 $\langle(d\exp_p)_v(v),(d\exp_p)_v(w_T)\rangle=\langle v,w_T\rangle$, it suffices
 to prove (2) for $w=w_N$ ($w_N\ne 0$). Pick a curve $v(s)$ in $T_pM$ with
-$v(0)=v$, $v'(0)=w_N$, $|v(s)|=\text{const}$, and consider the parametrized
-surface $f(t,s)=\exp_p tv(s)$ whose curves $t\mapsto f(t,s_0)$ are geodesics.
+$v(0)=v$, $v'(0)=w_N$, $|v(s)|=\text{const}$. Since $\exp_p v$ is defined,
+choose $\varepsilon>0$ so that $\exp_p(tv(s))$ is defined for $0\le t\le 1$,
+$|s|<\varepsilon$, and consider the parametrized surface
+$f(t,s)=\exp_p tv(s)$ whose curves $t\mapsto f(t,s_0)$ are geodesics.
 Then $\langle\frac{\partial f}{\partial s},\frac{\partial f}{\partial t}\rangle(1,0)=\langle(d\exp_p)_v(w_N),(d\exp_p)_v(v)\rangle$.
 Since $\frac{\partial f}{\partial t}$ is the tangent of a geodesic and by symmetry
 of the connection,
@@ -375,7 +384,8 @@ The isometries of a Riemannian manifold take geodesics into geodesics.
 ### 3.10 Example (the Lobatchevski plane)
 Let $G=\{(x,y)\in\mathbb{R}^2;\ y>0\}$ with metric $g_{11}=g_{22}=\frac{1}{y^2}$,
 $g_{12}=g_{21}=0$. The $y$-axis segment $\gamma(t)=(0,t)$, $a\le t\le b$ ($a>0$),
-is a geodesic: for any arc $c(t)=(x(t),y(t))$ with $c(a)=(0,a)$, $c(b)=(0,b)$,
+is the image of a geodesic: for any arc $c(t)=(x(t),y(t))$ with $c(a)=(0,a)$,
+$c(b)=(0,b)$,
 
 $$
 \ell(c)=\int_a^b\sqrt{(x')^2+(y')^2}\,\frac{dt}{y}\ge\int_a^b\frac{|y'|}{y}\,dt\ge\int_a^b\frac{dy}{y}=\ell(\gamma),
@@ -404,7 +414,10 @@ $q\in M$ to the geodesic sphere $S_r(p)$ of radius $r<c$ stays outside the
 geodesic ball $B_r(p)$ in some neighborhood of $q$.
 *Proof.* Let $W$ be a totally normal neighborhood of $p$; by the lemma of
 homogeneity restrict to unit-speed geodesics, so to the unit bundle
-$T_1W=\{(q,v);\ q\in W,\ v\in T_qM,\ |v|=1\}$. Let
+$T_1W=\{(q,v);\ q\in W,\ v\in T_qM,\ |v|=1\}$. After shrinking $W$, assume that
+$\overline{W}$ is compactly contained in a normal neighborhood $U$ of $p$; since
+$T_1\overline{W}$ is compact, decrease $I$ so that
+$\gamma(I\times T_1W)\subset U$. Let
 $\gamma:I\times T_1W\to M$, $I=(-\varepsilon,\varepsilon)$, be the differentiable
 map with $t\mapsto\gamma(t,q,v)$ the unit-speed geodesic through $q$ with velocity
 $v$ at $t=0$. Set $u(t,q,v)=\exp_p^{-1}(\gamma(t,q,v))$ and
@@ -430,7 +443,7 @@ $(0,q,v)$, so in a neighborhood of $q$ its points stay outside $B_r(p)$. $\squar
 For any $p\in M$ there exists $\beta>0$ such that the geodesic ball $B_\beta(p)$ is
 strongly convex.
 *Proof.* Let $c$ be as in Lemma 4.1. Choose $\delta>0$ and $W$ as in Theorem 3.7
-with $\delta<\frac{c}{2}$, and take $\beta<\delta$ with $B_\beta(p)\subset W$. Let
+with $\delta<\frac{c}{2}$, and take $\beta<\delta$ with $\overline{B_\beta(p)}\subset W$. Let
 $q_1,q_2\in\overline{B_\beta(p)}$ and let $\gamma$ be the unique geodesic of length
 $<2\delta<c$ joining them; then $\gamma\subset B_c(p)$. If the interior of
 $\gamma$ is not contained in $B_\beta(p)$, there is an interior point $m$ where the
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/04-curvature.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/04-curvature.mdx
index 23a5a64c..13268c76 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/04-curvature.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/04-curvature.mdx
@@ -473,7 +473,7 @@ $$
 for all $X,Y\in\mathcal{X}(M)$, $f,g\in\mathcal{D}(M)$.
 
 A tensor is a pointwise object. Fix $p\in M$ and let $U$ be a neighborhood of $p$
-on which it is possible to define vector fields $E_1,\dots,E_n\in\mathcal{X}(M^n)$
+on which it is possible to define vector fields $E_1,\dots,E_n\in\mathcal{X}(U)$
 such that at each $q\in U$ the vectors $\{E_i(q)\}$, $i=1,\dots,n$, form a basis of
 $T_qM$; we say $\{E_i\}$ is a *moving frame* on $U$. Writing
 $Y_1=\sum_{i_1}y_{i_1}E_{i_1},\dots,Y_r=\sum_{i_r}y_{i_r}E_{i_r}$, by linearity
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/05-jacobi.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/05-jacobi.mdx
index f13f7b80..f3ca911b 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/05-jacobi.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/05-jacobi.mdx
@@ -145,9 +145,10 @@ $\gamma(t)$ with $J(0)=0$.
 
 ### 2.4 Proposition
 Let $\gamma:[0,a]\to M$ be a geodesic and let $J$ be a Jacobi field along
-$\gamma$ with $J(0)=0$. Put $\frac{DJ}{dt}(0)=w$ and $\gamma'(0)=v$. Consider $w$
-as an element of $T_{av}(T_{\gamma(0)}M)$ and construct a curve $v(s)$ in
-$T_{\gamma(0)}M$ with $v(0)=av$, $v'(0)=w$. Put $f(t,s)=\exp_p(\frac{t}{a}v(s))$,
+$\gamma$ with $J(0)=0$. Put $\frac{DJ}{dt}(0)=w$ and $\gamma'(0)=v$. Using the
+natural identification $T_{av}(T_{\gamma(0)}M)\simeq T_{\gamma(0)}M$, construct a
+curve $v(s)$ in $T_{\gamma(0)}M$ with $v(0)=av$, $v'(0)=aw$. Put
+$f(t,s)=\exp_p(\frac{t}{a}v(s))$,
 $p=\gamma(0)$, and define a Jacobi field $\bar J$ by
 $\bar J(t)=\frac{\partial f}{\partial s}(t,0)$. Then $\bar J=J$ on $[0,a]$.
 
@@ -169,8 +170,8 @@ Since $J(0)=\bar J(0)=0$ and $\frac{DJ}{dt}(0)=\frac{D\bar J}{dt}(0)=w$, we
 conclude, from the uniqueness theorem, that $J=\bar J$. $\square$
 
 ### 2.5 Corollary
-Let $\gamma:[0,a]\to M$ be a geodesic. Then a Jacobi field $J$ along $\gamma$ with
-$J(0)=0$ is given by
+Let $\gamma:[0,a]\to M$ be a geodesic and put $p=\gamma(0)$. Then a Jacobi field
+$J$ along $\gamma$ with $J(0)=0$ is given by
 
 $$
 J(t)=(d\exp_p)_{t\gamma'(0)}(tJ'(0)),\quad t\in[0,a].
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/06-immersions.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/06-immersions.mdx
index f0179c75..e3967374 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/06-immersions.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/06-immersions.mdx
@@ -238,7 +238,7 @@ Using (2) and (3) we obtain (1). $\square$
 
 ### 2.6 Remark (Gauss formula for a hypersurface)
 In the case of a hypersurface $f:M^n\to\overline{M}^{n+1}$, let $p\in M$,
-$\eta\in(T_pM)^\perp$, and $\{e_1,\dots,e_n\}$ an orthonormal basis of $T_pM$ in
+$\eta\in(T_pM)^\perp$, $|\eta|=1$, and $\{e_1,\dots,e_n\}$ an orthonormal basis of $T_pM$ in
 which $S_\eta=S$ is diagonal, $S(e_i)=\lambda_ie_i$. Then $H(e_i,e_i)=\lambda_i$
 and $H(e_i,e_j)=0$ if $i\neq j$, so (1) becomes
 
@@ -302,7 +302,7 @@ $\sigma$, then $M$ has constant sectional curvature.
 
 ### 2.10 Definition (minimal immersion; mean curvature vector)
 An immersion $f:M\to\overline{M}$ is called *minimal* if for every $p\in M$ and
-every $\eta\in(T_pM)^\perp$ the trace of $S_\eta=0$. Choosing an orthonormal frame
+every $\eta\in(T_pM)^\perp$ the trace of $S_\eta$ is zero. Choosing an orthonormal frame
 $E_1,\dots,E_m$ of vectors in $\mathcal{X}(U)^\perp$, where $U$ is a neighborhood
 of $p$ in which $f$ is an embedding, we write at $p$
 
@@ -452,11 +452,12 @@ $$
 (\overline{\nabla}_X B)(Y,Z,\eta)=(\overline{\nabla}_Y B)(X,Z,\eta).
 $$
 
-If, in addition, the codimension of the immersion is $1$, then
+If, in addition, the codimension of the immersion is $1$ and $\eta$ is chosen as a
+local unit normal field, then
 $\nabla_X^\perp\eta=0$, hence
 
 $$
-\overline{\nabla}_X B(Y,Z,\eta)=X\langle S_\eta(Y),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle-\langle S_\eta(Y),\nabla_X Z\rangle=\langle\nabla_X(S_\eta(Y)),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle.
+(\overline{\nabla}_X B)(Y,Z,\eta)=X\langle S_\eta(Y),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle-\langle S_\eta(Y),\nabla_X Z\rangle=\langle\nabla_X(S_\eta(Y)),Z\rangle-\langle S_\eta(\nabla_X Y),Z\rangle.
 $$
 
 Therefore, in this case, the Codazzi equation can be written
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/07-hopf-rinow.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/07-hopf-rinow.mdx
index dbae423a..50556139 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/07-hopf-rinow.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/07-hopf-rinow.mdx
@@ -257,7 +257,7 @@ diffeomorphism.
 ### 3.2 Lemma (no conjugate points when $K\le 0$)
 Let $M$ be a complete Riemannian manifold with $K(p,\sigma)\le 0$ for all $p\in M$
 and for all $\sigma\subset T_pM$. Then for all $p\in M$ the conjugate locus
-$C(p)=\phi$; in particular the exponential map $\exp_p:T_pM\to M$ is a local
+$C(p)=\varnothing$; in particular the exponential map $\exp_p:T_pM\to M$ is a local
 diffeomorphism.
 *Proof.* Let $J$ be a non-trivial (that is, not identically zero) Jacobi field
 along a geodesic $\gamma:[0,\infty)\to M$, where $\gamma(0)=p$ and $J(0)=0$. Then
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/08-constant-curvature.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/08-constant-curvature.mdx
index 03442394..28167749 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/08-constant-curvature.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/08-constant-curvature.mdx
@@ -116,7 +116,7 @@ version was given by W. Ambrose in 1956.
 ### 2.2 Corollary
 Let $M$ and $\tilde{M}$ be spaces with the same constant curvature and the same
 dimension $n$. Let $p\in M$ and $\tilde{p}\in\tilde{M}$. Choose arbitrary
-orthonormal bases $\{e_j\}\in T_p(M)$ and $\{\tilde{e}_j\}\in T_{\tilde{p}}(\tilde{M})$,
+orthonormal bases $\{e_j\}$ of $T_p(M)$ and $\{\tilde{e}_j\}$ of $T_{\tilde{p}}(\tilde{M})$,
 $j=1,\dots,n$. Then there exist a neighborhood $V\subset M$ of $p$, a neighborhood
 $\tilde{V}\subset\tilde{M}$ of $\tilde{p}$, and an isometry $f:V\to\tilde{V}$ such
 that $df_p(e_j)=\tilde{e}_j$.
@@ -305,14 +305,18 @@ transformations.
 If $M$ is a Riemannian manifold and $\Gamma$ is a subgroup of the group of
 isometries acting in a totally discontinuous manner, then $M/\Gamma$ has a
 differentiable structure in which $\pi:M\to M/\Gamma$ is a local diffeomorphism,
-and a Riemannian metric making $\pi$ a local isometry, defined by
+and a Riemannian metric making $\pi$ a local isometry. Given $p\in M/\Gamma$,
+choose $\tilde p\in\pi^{-1}(p)$; for $u,v\in T_p(M/\Gamma)$, define
 
 $$
 \langle u,v\rangle=\langle d\pi^{-1}(u),d\pi^{-1}(v)\rangle_{\tilde{p}},
 $$
 
-called the *metric on $M/\Gamma$ induced by the covering $\pi$*. $M/\Gamma$ is
-complete iff $M$ is, and has constant curvature iff $M$ does. Taking $M=S^n$,
+Since $\pi$ is a regular covering, $\Gamma$ is transitive on $\pi^{-1}(p)$, and
+because $\Gamma$ consists of isometries the definition is independent of the
+choice of $\tilde p$. This is called the *metric on $M/\Gamma$ induced by the
+covering $\pi$*. $M/\Gamma$ is complete iff $M$ is, and has constant curvature iff
+$M$ does. Taking $M=S^n$,
 $\mathbb{R}^n$ or $H^n$, we conclude that $M/\Gamma$ is a complete manifold of
 constant curvature $1$, $0$ or $-1$, respectively.
 
@@ -379,10 +383,16 @@ The spaces of constant curvature have an important role in the historical
 development of Riemannian Geometry, due to their relationship with non-Euclidean
 geometry. A non-Euclidean geometry is a complete Riemannian manifold $M$ together
 with a transitive group of isometries $G$ (the non-Euclidean motions) satisfying
-the *Axiom of free mobility*. This implies that $M$ has constant sectional
-curvature, so the spaces of non-Euclidean Geometry are included among the space
-forms. The cases of $S^n$, $P^n$ and $H^n$ are called *spherical*, *elliptic* and
-*hyperbolic* geometry, respectively.
+the *Axiom of free mobility*: if $p,\tilde p\in M$, if $\gamma_1,\gamma_2$ are
+geodesics starting at $p$ and forming an angle $\alpha$, and if
+$\tilde\gamma_1,\tilde\gamma_2$ are geodesics starting at $\tilde p$ and forming
+the same angle $\alpha$, then there is a $g\in G$ with $g(p)=\tilde p$,
+$g(\gamma_1)=\tilde\gamma_1$, and $g(\gamma_2)=\tilde\gamma_2$. This corresponds
+to the side-angle-side condition for congruence of triangles in Euclidean geometry
+and implies that $M$ has constant sectional curvature, so the spaces of
+non-Euclidean Geometry are included among the space forms. The cases of $S^n$,
+$P^n$ and $H^n$ are called *spherical*, *elliptic* and *hyperbolic* geometry,
+respectively.
 
 ## §5 Isometries of the hyperbolic space; Theorem of Liouville
 
@@ -499,11 +509,12 @@ $$
 $$
 
 which we write as $1/\lambda=\rho=a_1|p-p_0|^2+k_1$ with $a_1=\sigma/2$,
-$k_1=\text{const.}$, $p_0\in\mathbb{R}^n$. Taking the inversion
+$k_1=\text{const.}$, $p_0\in\mathbb{R}^n$. The proof in this case will be complete
+once we show $k_1=0$: if $k_1=0$, taking the inversion
 $g(p)=\frac{p-p_0}{|p-p_0|^2}+p_0$ and $h=g\circ f^{-1}$, $h$ is conformal with
 coefficient $a_1$, hence an isometry followed by a dilatation; thus
 $f=h^{-1}\circ g$ is an inversion followed by a dilatation, followed by an
-isometry. To show $k_1=0$: applying the argument to $f^{-1}$,
+isometry. To show $k_1=0$, apply the argument to $f^{-1}$:
 $\lambda=a_2|f(p)-q_0|^2+k_2$, hence
 
 $$
@@ -519,7 +530,8 @@ $$
 $$
 
 If $k_1\neq 0$, the left side is a transcendental function of $|p(s_0)-p_0|$, while
-$(11)$ implies it is algebraic. This contradiction shows $k_1=0$.
+$(11)$ implies it is algebraic. This contradiction shows $k_1=0$, completing the
+case $\sigma\neq0$.
 
 *Case $\sigma=0$.* Here $\rho=1/\lambda=\sum a_i x_i+c_1=A_1(x)+c_1$. Applying the
 initial argument to $f^{-1}$,
@@ -551,16 +563,17 @@ onto $H^n$. Conversely, let $f:H^n\to H^n$ be a conformal transformation of $H^n
 onto $H^n$ and let $e_1,\dots,e_n$ be an orthonormal basis (in $g_{ij}$) at
 $p\in H^n$. Since $g_{ij}=\delta_{ij}/x_n^2$ and $f$ is conformal, there exists
 $\lambda^2>0$ with $\langle df_p(e_i),df_p(e_j)\rangle=\lambda^2\delta_{ij}$, so
-$\{df_p(e_i)/\lambda\}$ is orthonormal at $f(p)$. By Corollary 2.3 of Cartan's
-Theorem, there exists an isometry $g$ of $H^n$ taking $p$ to $f(p)$ with
-$dg(e_i)=df(e_i)/\lambda$. By the first part, $g$ is conformal. Hence
+$\{df_p(e_i)/\lambda\}$ is orthonormal at $f(p)$. Corollary 2.3 of Cartan's
+Theorem, together with the global construction after Theorem 2.1 (the exponential
+maps of $H^n$ are global diffeomorphisms), gives an isometry $g$ of $H^n$ taking
+$p$ to $f(p)$ with $dg(e_i)=df(e_i)/\lambda$. By the first part, $g$ is conformal. Hence
 $h=g^{-1}\circ f$ is the restriction to $H^n$ of a conformal map of $\mathbb{R}^n$
 taking $H^n$ onto $H^n$, leaving $p$ fixed and satisfying $dh_p=\lambda I$. It
 remains to show $h=\text{identity}$.
 
 Let $P$ be a hyperplane through $p$. By Liouville's theorem, $h(P)$ is a hyperplane
 or a sphere through $p$; since $dh_p$ is a multiple of the identity, $P$ and $h(P)$
-are tangent at $P$. Since $h$ takes $\partial H^n$ into itself and is conformal,
+are tangent at $p$. Since $h$ takes $\partial H^n$ into itself and is conformal,
 the angle of $P$ with $\partial H^n$ equals the angle of $h(P)$ with $\partial H^n$.
 We claim $h(P)=P$. Consider a line $r_1$ through $p$ perpendicular to
 $\partial H^n$, $q_1=r_1\cap\partial H^n$. Since $h(r_1)$ is a circle or a line
@@ -576,15 +589,18 @@ onto $H^n$, $h$ is the identity.
 For $n=2$, a simple calculation shows the conformal transformations of the form
 
 $$
-f(z)=\frac{az+b}{cz+d},\quad z\in H^2\subset\mathbb{C},\ a,b,c,d\in\mathbb{R},\ ad-bc\neq 0\qquad(12)
+f(z)=\frac{az+b}{cz+d},\quad z\in H^2\subset\mathbb{C},\ a,b,c,d\in\mathbb{R},\ ad-bc>0\qquad(12)
 $$
 
 (which map $H^2$ onto $H^2$) are isometries of $H^2$ with the metric $g_{ij}$.
 Moreover, for a fixed point $p_0\in H^2$ and unit vector $v_0$ at $p_0$, there
 exists a transformation $(12)$ taking an arbitrary $(p,v)$ to $(p_0,v_0)$ ($p$ and
 $v$ are determined by three parameters, the number of parameters of $f$). Since
-there is a unique isometry of $H^2$ taking $(p,v)$ to $(p_0,v_0)$, all isometries
-of $H^2$ are of the form $(12)$. $\square$
+there is a unique orientation-preserving isometry of $H^2$ taking $(p,v)$ to
+$(p_0,v_0)$, all orientation-preserving isometries of $H^2$ are of the form (12).
+Composing these with $z\mapsto-\bar z$ gives the orientation-reversing
+anti-holomorphic isometries, so together they give all isometries of $H^2$.
+$\square$
 
 To conclude this section, we identify some important hypersurfaces of $H^n$. The
 intersection with $H^n$ of hyperplanes of $\mathbb{R}^n$ orthogonal to $\partial H^n$,
@@ -608,9 +624,9 @@ since $\langle df_p(v),df_p(v)\rangle=16\langle v,v\rangle/|p-p_0|^4$ and
 $f_n(p)=(4-|p|^2)/|p-p_0|^2$ give
 $\langle df_p(v),df_p(v)\rangle/(f_n(p))^2=\langle v,v\rangle/(1-\tfrac14|p|^2)^2$.
 Because $f$ is injective, $f$ is an isometry of $B^n$ into $H^n$, taking
-$\partial B^n-\{p_0\}$ into $\partial H^n$. By Liouville's Theorem, $g:B^n\to B^n$
-is an isometry in $h_{ij}$ iff $g$ is the restriction of a conformal transformation
-of $\mathbb{R}^n$ taking $B^n$ onto $B^n$.
+$\partial B^n-\{p_0\}$ into $\partial H^n$. Conjugating by (13) and applying
+Theorem 5.3, $g:B^n\to B^n$ is an isometry in $h_{ij}$ iff $g$ is the restriction
+of a conformal transformation of $\mathbb{R}^n$ taking $B^n$ onto $B^n$.
 
 Let $S\subset H^n$ be an $(n-1)$-sphere of Euclidean space completely contained in
 $H^n$. Then $S$ is a *geodesic sphere* of $H^n$: its image $f^{-1}(S)\subset B^n$ is
@@ -625,15 +641,17 @@ metric on $P$ is a multiple of the Euclidean metric, so $P$ has constant curvatu
 zero, as does $S-\{p\}$. Such submanifolds are called *horospheres* of $H^n$.
 
 Let $S$ be a Euclidean sphere cutting $\partial H^n$ at an angle $\alpha$, and
-denote $S\cap H^n=\sum$. Through an inversion at a point of $S\cap\partial H^n$,
-$\sum$ is mapped isometrically into the intersection with $H^n$ of a hyperplane $P$
+denote $S\cap H^n=\Sigma$. Through an inversion at a point of $S\cap\partial H^n$,
+$\Sigma$ is mapped isometrically into the intersection with $H^n$ of a hyperplane $P$
 cutting $\partial H^n$ at the same angle $\alpha$. Let $Q$ be the hyperplane
 orthogonal to $\partial H^n$ containing $P\cap\partial H^n$. Then $P$ is a
 hypersurface equidistant from the totally geodesic hypersurface $Q$: for a geodesic
-$\gamma_r$ (semi-circle of radius $r$ with center $0\in P\cap\partial H^n$), a
+$\gamma_r$ (a semi-circle of radius $r$ with center $0\in P\cap\partial H^n$,
+lying in the plane perpendicular to $P\cap\partial H^n$), a
 homothety with center $0$ takes the circle of radius $r$ into one of any radius, so
 the length of $\gamma_r$ between its intersections with $P$ and $Q$ does not depend
-on $r$. We conclude that $P$, or its isometric image $\sum$, is obtained by taking
+on $r$. We conclude that $P$, or its isometric image $\Sigma$, is obtained by taking
 geodesics perpendicular to $Q$ and marking a fixed distance. Such hypersurfaces are
 called *equidistant surfaces* (or *hyperspheres*). In Exercise 6 it is shown that
-the hypersurfaces above are exactly the umbilic ones.
+the hypersurfaces above are exactly the umbilic ones, and their mean and sectional
+curvatures are calculated.
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/09-variations.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/09-variations.mdx
index 7c9386b2..cb621e79 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/09-variations.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/09-variations.mdx
@@ -40,6 +40,8 @@ $s\in(-\varepsilon,\varepsilon)$. If $f$ is differentiable, the variation is sai
 *differentiable*. For each $s\in(-\varepsilon,\varepsilon)$, the parametrized curve
 $f_s:[0,a]\to M$ given by $f_s(t)=f(s,t)$ is called a *curve in the variation*, so
 a variation determines a family $f_s(t)$ of neighboring curves of $f_0(t)=c(t)$.
+The variation is proper exactly when the curves in this family have the same
+initial point $c(0)$ and endpoint $c(a)$.
 The parametrized differentiable curve given by $f_t(s)=f(s,t)$, $t$ fixed, is a
 *transversal curve of the variation*. The velocity vector of a transversal curve
 at $s=0$, defined by $V(t)=\frac{\partial f}{\partial s}(0,t)$, is a (piecewise
@@ -157,7 +159,8 @@ every proper variation $f$ of $c$, we have $\frac{dE}{ds}(0)=0$.
 for $f$ proper, $V(0)=V(a)=0$, and all terms of (1) are zero. Conversely, suppose
 $\frac{dE}{ds}(0)=0$ for all proper variations. Let
 $V(t)=g(t)\frac{D}{dt}\frac{dc}{dt}$, where $g:[0,a]\to\mathbf{R}$ is piecewise
-differentiable with $g(t)>0$ if $t\ne t_i$ and $g(t_i)=0$. Constructing a variation
+differentiable with $g(t)>0$ if $t\ne t_i$ and $g(t_i)=0$, $i=0,\dots,k+1$.
+Constructing a variation
 with this $V(t)$,
 
 $$
@@ -247,7 +250,8 @@ $$
 $$
 
 ### 2.10 Remark (the index form)
-It is often convenient to write (5) using
+It is often convenient to write (5) using, on each interval where $V$ is
+differentiable,
 $\frac{d}{dt}\langle V,\frac{DV}{dt}\rangle=\langle V,\frac{D^2 V}{dt^2}\rangle+\langle\frac{DV}{dt},\frac{DV}{dt}\rangle$.
 Taking a geodesic $\gamma:[0,a]\to M$ and a partition
 $0=t_0<t_1<\cdots<t_{k+1}=a$, this gives
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/10-rauch.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/10-rauch.mdx
index a7880f75..0e5520f7 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/10-rauch.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/10-rauch.mdx
@@ -55,7 +55,7 @@ $$
 $$
 
 Sturm's Theorem asserts that if $f'(0)=\tilde f'(0)>0$, $\tilde f(t)\neq 0$ on
-$(0,\ell)$, and $\widetilde{K}(t)\geq K(t)$, then $\tilde f(t)\leq f(t)$.
+$(0,\ell]$, and $\widetilde{K}(t)\geq K(t)$, then $\tilde f(t)\leq f(t)$.
 
 In dimension higher than two the proof is much less simple; a presentation of
 the theorem was made for the first time in 1951 by Rauch. The proof presented in
@@ -376,6 +376,8 @@ $$
 \tilde J(t)=\frac{\sinh(t\sqrt{-b})}{\sinh(\ell\sqrt{-b})}\tilde w(t)\ (b<0),\qquad\tilde J(t)=\frac{t}{\ell}\tilde w(t)\ (b=0),
 $$
 
+where $\tilde w$ is the parallel vector field along $\tilde\gamma$ with
+$\tilde w(\ell)=\tilde v$. Thus
 one obtains from the proof of the Index Lemma
 
 $$
@@ -636,7 +638,7 @@ orthogonal complement of $\gamma'(0)$. We say that $\gamma$ is *focal point free
 at $(0,a]$ if there exists some $\varepsilon>0$ such that $\gamma$ has no focal
 points relative to the submanifold $\Sigma_\varepsilon=\exp_{\gamma(0)}(B_\varepsilon(0))$.
 
-Observe that since $\Sigma_\varepsilon$ is geodesic at $p$, any Jacobi field $J$
+Observe that since $\Sigma_\varepsilon$ is geodesic at $\gamma(0)$, any Jacobi field $J$
 along $\gamma$, with $J(0)\neq 0$ and $J'(0)=0$, automatically satisfies
 $S_{\gamma'(0)}(J(0))=0$.
 
@@ -664,7 +666,7 @@ geodesics, with the same velocity, and let $J$ and $\tilde J$ be Jacobi fields
 along $\gamma$ and $\tilde\gamma$, such that
 
 $$
-0=J'(0)=\tilde J'(0),\quad\langle J(0),\gamma'(0)\rangle=\langle J(0),\tilde\gamma'(0)\rangle,\quad|J(0)|=|\tilde J(0)|.
+0=J'(0)=\tilde J'(0),\quad\langle J(0),\gamma'(0)\rangle=\langle \tilde J(0),\tilde\gamma'(0)\rangle,\quad|J(0)|=|\tilde J(0)|.
 $$
 
 Assume that $\tilde\gamma$ is focal point free on $(0,a]$ and that, for all $t$
@@ -684,6 +686,12 @@ modifications: $\frac{|J|^2}{|\tilde J|^2}$ is well-defined, since $\tilde\gamma
 is focal point free, and $I_{t_o}(\phi(U),\phi(U))\geq I_{t_o}(\tilde U,\tilde U)$
 by Lemma 4.8. $\blacksquare$
 
+Other useful extensions of the Rauch comparison theorem can be found in F. Warner,
+"Extensions of the Rauch comparison theorem to submanifolds", Trans. A.M.S. 122
+(1966), 341-356, and in E. Heintze and H. Karcher, "A general comparison theorem
+with applications to volume estimates for submanifolds", Ann. Sci. Ecole Norm.
+Sup. 11 (1978), 451-470.
+
 Rauch's theorem admits an extremely important global generalization, called the
 Theorem of Toponogov. One of its versions: let $M$ be a Riemannian manifold which
 is complete with sectional curvature $K\geq H$. Let $\gamma_1$ and $\gamma_2$ be
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/11-morse.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/11-morse.mdx
index 67308f0a..4a49da10 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/11-morse.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/11-morse.mdx
@@ -285,7 +285,7 @@ But this is precisely the statement of the Index Theorem. $\square$
 Let $\gamma:[0,a]\to M$ be a geodesic segment on $M$ such that $\gamma(a)$ is not
 conjugate to $\gamma(0)$. Then $\gamma$ has no conjugate points on $(0,a)$ if and
 only if for all proper variations of $\gamma$ there exists a $\delta>0$ such that
-$E(s)<E(\delta)$ for $0<|s|<\delta$. In particular, if $\gamma$ is minimizing,
+$E(0)\leq E(s)$ for $|s|<\delta$. In particular, if $\gamma$ is minimizing,
 $\gamma$ has no conjugate points on $(0,a)$.
 
 ### 2.10 Corollary (conjugate points are discrete)
@@ -335,7 +335,7 @@ $T_w(\overset{\circ}{B})$ to a broken geodesic $w$ corresponds to the set of
 Jacobi fields along $w$, broken at $t_i$. It is possible to show that there
 exists a homotopy
 $h_s:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{\Omega}{}^c$, $s\in[0,1]$ with
-$h_o=\mathrm{ident.}$ and $h_1:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{B}$,
+$h_0=\mathrm{ident.}$ and $h_1:\overset{\circ}{\Omega}{}^c\to\overset{\circ}{B}$,
 that is, $\overset{\circ}{B}$ is a "deformation retract" of
 $\overset{\circ}{\Omega}{}^c$.
 
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/12-fundamental-group.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/12-fundamental-group.mdx
index 0e27bc4a..1740d2ba 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/12-fundamental-group.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/12-fundamental-group.mdx
@@ -156,13 +156,13 @@ of $\tilde{M}$, that is, if $f(c)=c$, where $c=\tilde{\gamma}((-\infty,\infty))$
 In this case, we say that $f$ is a *translation along* $\tilde{\gamma}$.
 
 ### 2.6 Proposition (covering transformations are translations)
-Let $M$ be a compact Riemannian manifold and $\alpha$ a covering transformation
+Let $M$ be a compact Riemannian manifold and let $\alpha\neq\mathrm{ident}$ be a covering transformation
 of $\tilde{M}$, considered with the covering metric. Then $\alpha$ is a
 translation of $\tilde{M}$.
 
 *Proof.* Let $\tilde{p}\in\tilde{M}$ and let $g\in\pi_1(M;p)$, $p=\pi(\tilde{p})$,
 the element corresponding to $\alpha$ under the isomorphism mentioned in the
-introduction of this Section. We can assume $\alpha\neq\mathrm{ident}$. By
+introduction of this Section. By
 Cartan's Theorem (2.2), there exists a closed geodesic $\gamma$ of $M$ in the
 free homotopy class determined by $g$. Choose a point $q\in\gamma$. Then $\gamma$
 is homotopic to the closed path $\sigma g\sigma^{-1}$, where $\sigma$ is a path
@@ -188,7 +188,7 @@ It follows that if $\tilde{\gamma}(s)$ is a point of the lift of $\gamma$ starti
 from $\tilde{q}$, we have, by uniqueness of lifting,
 
 $$
-\alpha(\tilde{\gamma}(s))=\alpha_{\tilde{q}}(\tilde{\gamma}(s))\in\gamma,
+\alpha(\tilde{\gamma}(s))=\alpha_{\tilde{q}}(\tilde{\gamma}(s))\in\tilde{\gamma},
 $$
 
 which shows that $\tilde{\gamma}$ is invariant by $\alpha$, and proves the
@@ -245,16 +245,18 @@ Since $K\leq 0$ and $T_c(\tilde{M})$ has zero curvature, we can apply Propositio
 2.5 of Chapter 10 (application of Rauch's Theorem) and obtain that
 
 $$
-\ell(\Gamma_C)\leq\ell(\gamma_C)\quad(<,\text{ if }K<0).
+\ell(\Gamma_C)\leq\ell(\gamma_C),
 $$
 
+with strict inequality if $K<0$.
 It follows that
 
 $$
-A^2+B^2-2AB\cos\gamma\leq\ell(\Gamma_C)^2\leq\ell(\gamma_C)^2=C^2\quad(<,\text{ if }K<0),
+A^2+B^2-2AB\cos\gamma\leq\ell(\Gamma_C)^2\leq\ell(\gamma_C)^2=C^2,
 $$
 
-which proves (i).
+where the last inequality is strict if $K<0$.
+This proves (i).
 
 To prove (ii), let us observe that
 
@@ -268,9 +270,10 @@ sides have lengths $A$, $B$ and $C$. Denoting the opposite angles of this triang
 by $\alpha'$, $\beta'$ and $\gamma'$, respectively, we obtain from (i),
 
 $$
-\alpha\leq\alpha',\qquad\beta\leq\beta',\qquad\gamma\leq\gamma'\quad(<,\text{ if }K<0).
+\alpha\leq\alpha',\qquad\beta\leq\beta',\qquad\gamma\leq\gamma'.
 $$
 
+All three inequalities are strict if $K<0$.
 Since $\alpha'+\beta'+\gamma'=\pi$, (ii) follows. $\blacksquare$
 
 From now on, $M$ will denote a complete Riemannian manifold with sectional
@@ -404,17 +407,20 @@ If $M$ is compact, $K<0$, and $H$ is a solvable subgroup of $\pi_1(M)$,
 $H\neq\{e\}$, then $H$ is infinite cyclic. In addition, $\pi_1(M)$ does not have a
 cyclic subgroup of finite index.
 
-*Proof.* Since $H$ is solvable, there exists a finite sequence of subgroups
+*Proof.* Since $H$ is solvable, choose a finite sequence of subgroups with $k$
+minimal:
 
 $$
-H=H_o\supset H_1\supset\cdots\supset H_{k-1}\supset H_k=\{e\}
+H=H_0\supset H_1\supset\cdots\supset H_{k-1}\supset H_k=\{e\}
 $$
 
 such that $H_{i+1}$ is normal in $H_i$ and $H_i/H_{i+1}$ is abelian. Then
-$H_{k-1}$ is abelian, hence, by Preissman's Theorem (3.2), infinite cyclic.
+$H_{k-1}\neq\{e\}$ and $H_{k-1}$ is abelian, hence, by Preissman's Theorem (3.2),
+infinite cyclic. If $k=1$, this proves the first assertion. Assume $k\geq2$.
 
 Let $g\in\pi_1(M)$ be a generator of $H_{k-1}$ and let $\tilde{\gamma}$ be the
-geodesic in $\tilde{M}$ invariant under $g$. Let $a\in H_{k-2}$ and $b\in H_{k-1}$.
+geodesic in $\tilde{M}$ invariant under $g$. Let $a\in H_{k-2}$, and choose a
+nonidentity element $b\in H_{k-1}$, for instance $b=g$.
 Since $a^{-1}b^{-1}ab\in H_{k-1}$, we have, for some integer $n$,
 
 $$
@@ -437,16 +443,21 @@ Repeating the argument above a finite number of times, we conclude that $H$ is
 infinite cyclic, which proves the first statement of the theorem.
 
 To prove the final assertion, suppose that there exists a subgroup
-$H\subset\pi_1(M)$, cyclic and of finite index. Let $g$ be a generator of $H$ and
-let $\tilde{\gamma}$ be the geodesic of $\tilde{M}$ invariant under $g$. Let
-$a\in\pi_1(M)-H$. Since $H$ has finite index, there exist integers $m$ and $n$
-such that $a^n=g^m$. Therefore
+$H\subset\pi_1(M)$, cyclic and of finite index. If $H=\{e\}$, then $\pi_1(M)$ is
+finite; this contradicts Preissman's Theorem (3.2) unless $\pi_1(M)=\{e\}$, which
+contradicts Theorem 3.8. Thus $H=\langle g\rangle$ with $g\neq e$. If
+$H=\pi_1(M)$, then $\pi_1(M)$ is abelian, again contradicting Theorem 3.8. Hence
+$H$ is proper. Let $\tilde{\gamma}$ be the geodesic of $\tilde{M}$ invariant under
+$g$, and let $a\in\pi_1(M)-H$. Since $H$ has finite index, choose $n>0$ with
+$a^n\in H$. Applying Preissman's Theorem to the subgroup generated by $a$, we have
+$a^n\neq e$, so $a^n=g^m$ for some $m\neq0$. Therefore
 
 $$
 a^n(\tilde{\gamma})=g^m(\tilde{\gamma})=\tilde{\gamma}.
 $$
 
-By uniqueness, $a(\tilde{\gamma})=\tilde{\gamma}$, for all $a\in\pi_1(M)-H$. It
+The nonidentity transformation $a^n$ also leaves $a(\tilde{\gamma})$ invariant, so
+by uniqueness, $a(\tilde{\gamma})=\tilde{\gamma}$, for all $a\in\pi_1(M)-H$. It
 follows that every element of $\pi_1(M)$ leaves invariant the geodesic
 $\tilde{\gamma}$. By Lemma 3.5, $\pi_1(M)$ is infinite cyclic, which contradicts
 Theorem 3.8. $\blacksquare$
@@ -454,8 +465,9 @@ Theorem 3.8. $\blacksquare$
 ### 3.11 Remark
 Riemannian manifolds of non-positive curvature form a substantial topic in
 Riemannian Geometry, which we have barely touched. To the reader interested in
-some of the recent developments, we recommend the survey by P. Eberlein,
+some of the recent developments, we recommend the excellent survey by P. Eberlein,
 "Structure of manifolds of nonpositive curvature", in *Global Geometry and Global
-Analysis 1984*, Proceedings, Berlin, Lecture Notes in Math. 1156, Springer Verlag,
-1985, pp. 86–153. Other relationships between the geometry of a manifold $M$ of
+Analysis 1984*, Proceedings, Berlin, edited by D. Ferus, R. Gardner, S. Helgason
+and U. Simon, Lecture Notes in Math. 1156, Springer Verlag, 1985, pp. 86–153.
+Other relationships between the geometry of a manifold $M$ of
 non-positive curvature and $\pi_1(M)$ are described in Section 7 of this survey.
diff --git a/projects/riemannian-geometry/.astrolabe/docs-src/13-sphere-theorem.mdx b/projects/riemannian-geometry/.astrolabe/docs-src/13-sphere-theorem.mdx
index 2cceb276..d49306ac 100644
--- a/projects/riemannian-geometry/.astrolabe/docs-src/13-sphere-theorem.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs-src/13-sphere-theorem.mdx
@@ -238,7 +238,7 @@ $0<K_{\min}\leq K\leq K_{\max}$, then:
 
 *Proof.* By Bonnet–Myers (Theor. 3.1 of Chap. 9), $M$ is compact. Since $T_1M$ is
 compact, Proposition 2.9 gives $p\in M$ with $d(p,C_m(p))=\inf_{r\in M}d(r,C_m(r))$,
-and $C_m(p)$ compact gives $q\in C_m(p)$ assuming this distance. If $q$ is conjugate
+and $C_m(p)$ compact gives $q\in C_m(p)$ realizing this distance. If $q$ is conjugate
 to $p$, then $d(p,q)\geq\pi/\sqrt{K_{\max}}$ by Proposition 2.4 of Chapter 10 — case
 (a). If $q$ is not conjugate to $p$, by Proposition 2.12 there are two minimizing
 geodesics $\mu,\sigma$ from $p$ to $q$ with $\mu'(\ell)=-\sigma'(\ell)$,
@@ -271,6 +271,26 @@ $|W(t)|=|\bar W(t)|$, $\langle W',W'\rangle=\langle\bar W',\bar W'\rangle$. Sinc
 $\sup\bar K\leq\inf K$, $I(W,W)\leq I(\bar W,\bar W)$; so $I(\bar W,\bar W)<0$ forces
 $I(W,W)<0$. $\square$
 
+For the next lemma we need the following facts. Let $f:M^n\to\mathbb{R}$ be a
+differentiable function on a differentiable manifold $M$. A point $p\in M$ is a
+critical point of $f$ if $df(p)=0$; $f(p)$ is then called a critical value of $f$.
+If $p$ is a critical point and $(x_1,\dots,x_n)$ is a coordinate system on $M$
+around $p$, the hessian of $f$, defined by the matrix
+$(\partial^2f/\partial x_i\partial x_j)(p)$, represents a symmetric bilinear form
+on $T_pM$ that does not depend on the chosen system of coordinates. We say that
+the critical point is non-degenerate if the determinant of this matrix is
+different from zero. Such a critical point is isolated, and it is possible to
+choose a coordinate system $(x_1,\dots,x_{n-\lambda},y_1,\dots,y_\lambda)$ in such
+a way that, in the coordinate neighborhood considered,
+
+$$
+f(x_1,\dots,x_{n-\lambda},y_1,\dots,y_\lambda)
+=f(p)+x_1^2+\cdots+x_{n-\lambda}^2-y_1^2-\cdots-y_\lambda^2.
+$$
+
+The integer $\lambda$ is called the index of the non-degenerate critical point
+$p$. (For more details see Milnor [Mi].)
+
 ### 3.3 Lemma (Morse-theoretic deformation)
 Let $M^n$ be a differentiable manifold and $f:M^n\to\mathbb{R}$ a function whose
 critical points are all non-degenerate. Given $p,q\in M$ and a differentiable curve
@@ -292,8 +312,9 @@ $\bar\gamma_1$ off $L$ within $H$ fixing endpoints; the gradient flow then retra
 into $M^{c_1-\varepsilon}$. Proceeding inductively yields
 $\bar\gamma\subset M^{a+\delta}$. $\square$
 
-(Remark after Lemma 3.3: if instead the largest critical value $c$ in $f^{-1}([a,b])$
-has index zero or one, then $\gamma$ is homotopic to $\bar\gamma$ with
+(Remark after Lemma 3.3: if there are critical points of index zero or one in
+$f^{-1}([a,b])$ and $c$ is the largest value of such critical points, then
+$\gamma$ is homotopic to $\bar\gamma$ with
 $\bar\gamma([0,1])\subset M^{c+\delta}$.)
 
 *Proof of Proposition 3.1.* Suppose $i(M)<\pi$. By Proposition 2.13 there is a
@@ -388,7 +409,7 @@ M=B_\rho(p)\cup B_\rho(q),
 $$
 
 where $B_r(p)$ denotes the open geodesic ball of radius $r$ and center $p$, and
-$\rho$ is such that $\pi/2\sqrt\delta<\rho<\pi$.
+$\rho$ is such that $\pi/(2\sqrt\delta)<\rho<\pi$.
 
 *Proof.* By the injectivity-radius estimate of Section 3, $B_\rho(p)$ contains no
 points of $C_m(p)$, so it is diffeomorphic via $\exp$ to a Euclidean ball, and
@@ -405,14 +426,14 @@ $r_o$; by Berger's Lemma there is $\gamma$ minimizing from $p$ to $q$ with
 $\langle\gamma'(0),\lambda'(0)\rangle\geq0$. Let $s\in\gamma$ with $d(p,s)=\rho$.
 Applying Rauch's Theorem (Proposition 2.5 of Chap. 10), comparing $M^n$ with a
 sphere $S^n$ of curvature $\delta$: since the angle
-$\sphericalangle r_o p s\leq\pi/2$, $d(r_o,s)\leq\pi/2\sqrt\delta$. As
+$\sphericalangle r_o p s\leq\pi/2$, $d(r_o,s)\leq\pi/(2\sqrt\delta)$. As
 $d(r_o,p)=d(r_o,q)=\rho$ and some $s$ has $d(r_o,s)<\rho$, the distance from $r_o$ to
 $\gamma$ is realized by an interior point $s_o$, with the minimizing geodesic from
 $r_o$ to $s_o$ orthogonal to $\gamma$ and
-$d(r_o,\gamma)=d(r_o,s_o)\leq\pi/2\sqrt\delta$. Since $d(p,q)\leq\pi/\sqrt\delta$,
-either $d(p,s_o)\leq\pi/2\sqrt\delta$ or $d(q,s_o)\leq\pi/2\sqrt\delta$; in the
-former case, since $d(r_o,s_o)\leq\pi/2\sqrt\delta$ and
-$\sphericalangle p s_o r_o=\pi/2$, Rauch gives $d(p,r_o)\leq\pi/2\sqrt\delta<\rho$,
+$d(r_o,\gamma)=d(r_o,s_o)\leq\pi/(2\sqrt\delta)$. Since $d(p,q)\leq\pi/\sqrt\delta$,
+either $d(p,s_o)\leq\pi/(2\sqrt\delta)$ or $d(q,s_o)\leq\pi/(2\sqrt\delta)$; in the
+former case, since $d(r_o,s_o)\leq\pi/(2\sqrt\delta)$ and
+$\sphericalangle p s_o r_o=\pi/2$, Rauch gives $d(p,r_o)\leq\pi/(2\sqrt\delta)<\rho$,
 contradicting $d(p,r_o)=\rho$ (the other case is analogous). $\square$
 
 ### 4.3 Lemma (the equator)
diff --git a/projects/riemannian-geometry/.astrolabe/docs/01-metrics.mdx b/projects/riemannian-geometry/.astrolabe/docs/01-metrics.mdx
index 787dbb3e..6fef0f7f 100644
--- a/projects/riemannian-geometry/.astrolabe/docs/01-metrics.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs/01-metrics.mdx
@@ -44,10 +44,6 @@ and $M^n=M$ has dimension $n$.
 
 \entryblock{a9965ea4095e}
 
-\entryblock{4a6501c59c3e}
-
-\entryblock{1934358d851d}
-
 ### Volume on an oriented manifold (between 2.10 and 2.11)
 For a positive parametrization $\mathbf{x}:U\to M$ about $p$, take a positive
 orthonormal basis $\{e_1,\dots,e_n\}$ of $T_pM$ and write
@@ -62,7 +58,7 @@ $$
 For another positive parametrization $\mathbf{y}:V\to M$ with $h_{ij}=\langle Y_i,Y_j\rangle$,
 
 $$
-\sqrt{\det(g_{ij})}(p)=J\sqrt{\det(h_{ij})}(p),\qquad J=\det\big(\tfrac{\partial x_i}{\partial y_j}\big)>0.\qquad\text{(4)}
+\sqrt{\det(g_{ij})}(p)=J\sqrt{\det(h_{ij})}(p),\qquad J=\det\big(\tfrac{\partial y_i}{\partial x_j}\big)>0.\qquad\text{(4)}
 $$
 
 For a region $R\subset M$ (open, connected, compact closure) contained in a
@@ -75,3 +71,7 @@ $$
 
 By the change-of-variables theorem and (4) this is independent of the chart
 (orientability guarantees $\mathrm{vol}(R)$ does not change sign).
+
+\entryblock{4a6501c59c3e}
+
+\entryblock{1934358d851d}
diff --git a/projects/riemannian-geometry/.astrolabe/docs/03-geodesics.mdx b/projects/riemannian-geometry/.astrolabe/docs/03-geodesics.mdx
index a7da14a9..89bd91d8 100644
--- a/projects/riemannian-geometry/.astrolabe/docs/03-geodesics.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs/03-geodesics.mdx
@@ -46,7 +46,7 @@ $TM$ is the set of pairs $(q,v)$, $q\in M$, $v\in T_qM$. If $(U,\mathbf{x})$ is
 coordinate system on $M$, any $v\in T_qM$, $q\in\mathbf{x}(U)$, is
 $\sum_i y_i\frac{\partial}{\partial x_i}$, and $(x_1,\dots,x_n,y_1,\dots,y_n)$ are
 coordinates of $(q,v)$ on $TU$, giving $TM$ a differentiable structure (cf.
-\entryref{317503003357} of Chap. 0). Locally $TU=U\times\mathbb{R}^n$, and the canonical
+\entryref{5c7d4e32a0d7} of Chap. 0). Locally $TU=U\times\mathbb{R}^n$, and the canonical
 projection $\pi:TM\to M$, $\pi(q,v)=q$, is differentiable. A curve $t\mapsto\gamma(t)$
 in $M$ determines $t\mapsto(\gamma(t),\frac{d\gamma}{dt}(t))$ in $TM$; if $\gamma$
 is a geodesic, this curve satisfies the first-order system
@@ -142,4 +142,3 @@ chosen so the ball is strongly convex.
 \entryblock{317503003357}
 
 \entryblock{49a53e470b68}
-
diff --git a/projects/riemannian-geometry/.astrolabe/docs/10-rauch.mdx b/projects/riemannian-geometry/.astrolabe/docs/10-rauch.mdx
index 0c9a907b..dd67cd06 100644
--- a/projects/riemannian-geometry/.astrolabe/docs/10-rauch.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs/10-rauch.mdx
@@ -55,7 +55,7 @@ $$
 $$
 
 Sturm's Theorem asserts that if $f'(0)=\tilde f'(0)>0$, $\tilde f(t)\neq 0$ on
-$(0,\ell)$, and $\widetilde{K}(t)\geq K(t)$, then $\tilde f(t)\leq f(t)$.
+$(0,\ell]$, and $\widetilde{K}(t)\geq K(t)$, then $\tilde f(t)\leq f(t)$.
 
 In dimension higher than two the proof is much less simple; a presentation of
 the theorem was made for the first time in 1951 by Rauch. The proof presented in
@@ -141,4 +141,3 @@ $\gamma$.
 \entryblock{cdfac76c48af}
 
 \entryblock{8286f921d5a2}
-
diff --git a/projects/riemannian-geometry/.astrolabe/docs/13-sphere-theorem.mdx b/projects/riemannian-geometry/.astrolabe/docs/13-sphere-theorem.mdx
index bc1342ec..29c51084 100644
--- a/projects/riemannian-geometry/.astrolabe/docs/13-sphere-theorem.mdx
+++ b/projects/riemannian-geometry/.astrolabe/docs/13-sphere-theorem.mdx
@@ -67,6 +67,26 @@ pass to \entryref{096001bed96a}.
 
 \entryblock{d11a88e897eb}
 
+For the next lemma we need the following facts. Let $f:M^n\to\mathbb{R}$ be a
+differentiable function on a differentiable manifold $M$. A point $p\in M$ is a
+critical point of $f$ if $df(p)=0$; $f(p)$ is then called a critical value of $f$.
+If $p$ is a critical point and $(x_1,\dots,x_n)$ is a coordinate system on $M$
+around $p$, the hessian of $f$, defined by the matrix
+$(\partial^2f/\partial x_i\partial x_j)(p)$, represents a symmetric bilinear form
+on $T_pM$ that does not depend on the chosen system of coordinates. We say that
+the critical point is non-degenerate if the determinant of this matrix is
+different from zero. Such a critical point is isolated, and it is possible to
+choose a coordinate system $(x_1,\dots,x_{n-\lambda},y_1,\dots,y_\lambda)$ in such
+a way that, in the coordinate neighborhood considered,
+
+$$
+f(x_1,\dots,x_{n-\lambda},y_1,\dots,y_\lambda)
+=f(p)+x_1^2+\cdots+x_{n-\lambda}^2-y_1^2-\cdots-y_\lambda^2.
+$$
+
+The integer $\lambda$ is called the index of the non-degenerate critical point
+$p$. (For more details see Milnor [Mi].)
+
 \entryblock{ff0d95de92e4}
 
 \entryblock{096001bed96a}