From 93c436ecd633e2f8539aa827d93964d5d5f63dc2 Mon Sep 17 00:00:00 2001
From: kaesle <kaesle@gmail.com>
Date: Sat, 28 Dec 2019 18:24:08 -0600
Subject: [PATCH 1/5] Typo fixed in definition right G-action

Satisfying (i), require p<|e=p instead of p<|g=p.
---
 ga/19principal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ga/19principal.tex b/ga/19principal.tex
index f02e481..6f4042f 100644
--- a/ga/19principal.tex
+++ b/ga/19principal.tex
@@ -49,7 +49,7 @@ \subsection{Lie group actions on a manifold}
 \ei
 satisfying
 \ben[label=\roman*)]
-\item $\forall \, p \in M :\ p\racts g = p$;
+\item $\forall \, p \in M :\ p\racts e = p$;
 \item $\forall \, g_1,g_2\in G : \forall \, p\in M : \ p \racts (g_1\bullet g_2) = (p \racts g_1) \racts g_2$.
 \een
 \ed

From 55367a2d09d3d1d90a77bd768acc58d872c24b57 Mon Sep 17 00:00:00 2001
From: kaesle <kaesle@gmail.com>
Date: Sat, 28 Dec 2019 18:36:29 -0600
Subject: [PATCH 2/5] Typo fixed in definition of transitive left G-action

Require q=g|>p for some g rather than p=g|>p.
---
 ga/19principal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ga/19principal.tex b/ga/19principal.tex
index 6f4042f..82b9a2b 100644
--- a/ga/19principal.tex
+++ b/ga/19principal.tex
@@ -197,7 +197,7 @@ \subsection{Lie group actions on a manifold}
 A left $G$-action $\lacts\cl G\times M\to M$ is said to be
 \ben[label=\roman*)]
 \item \emph{free} if for all $p\in M$, we have $S_p=\{e\}$;
-\item \emph{transitive} if for all $p,q\in M$, there exists $g\in G$ such that $p=g\lacts p$.
+\item \emph{transitive} if for all $p,q\in M$, there exists $g\in G$ such that $q=g\lacts p$.
 \een
 \ed
 

From a2b648a6c4eae89b807744c1a652d1beb8501fac Mon Sep 17 00:00:00 2001
From: kaesle <kaesle@gmail.com>
Date: Mon, 30 Dec 2019 09:17:52 -0600
Subject: [PATCH 3/5] Typo fixed in Proposition 19.14.

p element of base manifold M rather than of group G.
---
 ga/19principal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ga/19principal.tex b/ga/19principal.tex
index 82b9a2b..3fae340 100644
--- a/ga/19principal.tex
+++ b/ga/19principal.tex
@@ -229,7 +229,7 @@ \subsection{Lie group actions on a manifold}
 \bp
 If $\lacts\cl G \times M \to M$ is a free action, then
 \bse
-\forall \, p \in G :\ G_p \cong_{\mathrm{diff}} G.
+\forall \, p \in M :\ G_p \cong_{\mathrm{diff}} G.
 \ese
 \ep
 

From da2d7d5889cf5378295ba4655886c5c9b1b79c88 Mon Sep 17 00:00:00 2001
From: kaesle <kaesle@gmail.com>
Date: Mon, 30 Dec 2019 09:18:54 -0600
Subject: [PATCH 4/5] Typos fixed in definition of frame bundle

L_pM isomorphic as a Lie Group, not as vector space, to GL(d,R).
---
 ga/19principal.tex | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/ga/19principal.tex b/ga/19principal.tex
index 3fae340..06dec0a 100644
--- a/ga/19principal.tex
+++ b/ga/19principal.tex
@@ -281,11 +281,11 @@ \subsection{Principal fibre bundles}
 \ben[label=\alph*)]
 \item Let $M$ be a smooth manifold. Consider the space
 \bse
-L_pM := \{(e_1,\ldots,e_{\dim M})\mid e_1,\ldots,e_{\dim M} \text{ is a basis of }T_pM\} \cong_{\mathrm{vec}} \GL(\dim M,\R).
+L_pM := \{(e_1,\ldots,e_{\dim M})\mid e_1,\ldots,e_{\dim M} \text{ is a basis of }T_pM\} \cong_{\mathrm{LG}} \GL(\dim M,\R).
 \ese
 We know from linear algebra that the bases of a vector space are related to each other by invertible linear transformations. Hence, we have 
 \bse
-L_pM \cong_{\mathrm{vec}} \GL(\dim M,\R).
+L_pM \cong_{\mathrm{LG}} \GL(\dim M,\R).
 \ese
 We define the frame bundle of $M$ as
 \bse

From fc998e4ee7ad1eea1ea6734a1049541d2e54f44e Mon Sep 17 00:00:00 2001
From: kaesle <kaesle@gmail.com>
Date: Mon, 30 Dec 2019 09:25:40 -0600
Subject: [PATCH 5/5] Typo fixed in definition of frame bundle

T_pM -> L_pM
---
 ga/19principal.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/ga/19principal.tex b/ga/19principal.tex
index 06dec0a..d2d40fd 100644
--- a/ga/19principal.tex
+++ b/ga/19principal.tex
@@ -294,7 +294,7 @@ \subsection{Principal fibre bundles}
 with the obvious projection map $\pi\cl LM \to M$ sending each basis $(e_1,\ldots,e_{\dim M})$ to the unique point $p\in M$ such that $(e_1,\ldots,e_{\dim M})$ is a basis of $T_pM$.
 By proceeding similarly to the case of the tangent bundle, we can equip $LM$ with a smooth structure inherited from that of $M$. We then find
 \bse
-\dim LM = \dim M + \dim T_pM = \dim M + (\dim M)^2.
+\dim LM = \dim M + \dim L_pM = \dim M + (\dim M)^2.
 \ese
 \item We would now like to make $LM \xrightarrow{\,\pi\,}M$ into a principal $\GL(\dim M,\R)$-bundle. We define a right $\GL(\dim M,\R)$-action on $LM$ by
 \bse