Clarify cotangent lift

Signed-off-by: Marcello Seri <marcello.seri@gmail.com>

Author: mseri

6-cotangentbdl.tex

@@ -229,7 +229,7 @@ \section{Change of coordinates}
 	The \emph{pullback} of covectors by $F$ at the point $F(p)$, is the dual linear map of the differential
 	\marginnote[-1.5em]{Which is also the reason why, some authors, use the notation $F_*$ to denote the differential of maps between manifolds and other call pushforward the differential.}
 	\begin{equation}
-		dF^*_p : T^*_{F(p)} N \to T^*_p M, \quad \omega \mapsto dF^*\omega,
+		dF^*_p : T^*_{F(p)} N \to T^*_p M, \quad \omega \mapsto dF^*_p\omega,
 	\end{equation}
 	defined by duality in the following way\footnote{Or, omitting the point of application, $\left(dF^*\omega, v\right) := \left(\omega, dF(v)\right)$.}:
 	\begin{equation}
@@ -335,11 +335,11 @@ \section{One-forms and the cotangent bundle}
 	\end{equation}
 	defined by
 	\begin{equation}
-		(F^*\omega)_p := dF_p^*(\omega_{F(p)}).
+		(F^*\omega)_p := dF_p^*(\omega_{F(p)}), \quad \forall\, p\in M, \; \forall \omega\in\fX^*(N).
 	\end{equation}
 	By definition, this acts on vectors $v\in T_p M$ by
 	\begin{equation}
-		((F^*\omega)_p, v) = (\omega_{F(p)}, dF_p(v)) = \omega_{F(p)}(dF_p(v)).
+		((F^*\omega)_p \,|\, v) = (\omega_{F(p)} \,|\, dF_p(v)) = \omega_{F(p)}(dF_p(v)).
 	\end{equation}
 \end{definition}
 
@@ -365,7 +365,7 @@ \section{One-forms and the cotangent bundle}
 	For $p\in M$, denote $(V, (y^i))$ a chart on $N$ around $F(p)$ and let $U=F^{-1}(N)$.
 	If $\omega = \omega_j dy^j \in\fX^*(N)$, apply twice Exercise~\ref{ex:propdiff} to show that in $U$
 	\begin{equation}
-		F^*\omega = (\omega_j\circ F) d(y^j \circ F).
+		F^*\omega = (\omega_j\circ F)\, d(y^j \circ F).
 	\end{equation}
 
 	Let $F:\R^3\to\R^2$ be the map $(u,v) = F(x,y,z) = (x y^2, y \sin z)$.

7-tensors.tex

@@ -460,28 +460,29 @@ \section{Tensor bundles}
 \end{proposition}
 \begin{proof}
 	\newthought{Step I}.
-	We know that the pullback induces on the fibres a diffeomorphism of cotangent bundles. Let $p\in M$.
-	We have already seen that on the fibres the pullback is a diffeomorphism:
+	We know that the pullback induces on the fibres a diffeomorphism of cotangent bundles, obtained by collecting the pullbacks on each fibre into a single map:
 	\begin{equation}
 		T^*N \to T^* M, \quad
 		(q,\omega) \mapsto (\varphi^*\omega)_q = \left(\varphi^{-1}(q), d\varphi^*\omega|_{\varphi^{-1}(q)}\right).
 	\end{equation}
-	This can be inverted giving rise to the so-called \emph{cotangent lift}\footnote{This is often denoted also $\varphi_\sharp$.}
+	Since $\phi$ is a differential, this can be inverted giving rise to the so-called \emph{cotangent lift}\footnote{This is often denoted also $\varphi_\sharp$.} of $\varphi$:
 	\begin{equation}
-		d\varphi^\dagger :=(d\varphi)^\dagger := (\varphi^{-1})^*: T^*M \to T^*N.
+		d\varphi^\dagger: T^*M \to T^*N.
 	\end{equation}
+	This is a map that acts on each covector, pointwise, as $d(\varphi^{-1})^*$.%
 	\marginnote{An aid to understand this map is the following commuting diagram:
 		\begin{equation}\nonumber
 			\begin{tikzcd}[row sep=normal, column sep=normal, ampersand replacement=\&]
 				T_p^* M \arrow[d, "\pi_M" left] \arrow[r, "d\varphi^\dagger"] \& T^*_{\varphi(p)}N \arrow[d, "\pi_N"] \\
 				M \arrow[r, "\varphi" below] \& N
 			\end{tikzcd}
-		\end{equation}}
-	For any $\omega\in T_p^* M$ and any $v\in T_pM$, we have
+		\end{equation}}%
+	Note that if $p\in M$, for any $\omega\in T_p^* M$ and any $v\in T_pM$, we have
 	\begin{align}
-		(d\varphi^\dagger_p \omega \;\mid\; d\varphi_p v)_{\varphi(p)} & =  d(\varphi^{-1})^*(\omega|_{\varphi^{-1}\circ\varphi(p)} (d\varphi_p v) ) \\
-		                                                               & = \omega_p(d\varphi^{-1}_{\varphi(p)} \circ d\varphi_p v )                  \\
-		                                                               & = \omega_p(v) = (\omega \mid v)_p.
+		(d\varphi^\dagger_p \omega \;\mid\; d\varphi_p v)
+		& =  d(\varphi^{-1})^*(\omega|_{\varphi^{-1}\circ\varphi(p)} (d\varphi_p v) ) \\
+		& = \omega_p(d\varphi^{-1}_{\varphi(p)} \circ d\varphi_p v ) \\
+		& = \omega_p(v) = (\omega_p \mid v).
 	\end{align}
 
 	\newthought{Step II}.

aom.tex

@@ -294,7 +294,7 @@ \chapter*{Introduction}
 
 I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his invaluable suggestions, comments and corrections, and to Bram Brongers\footnote{You can also have a look at \href{https://fse.studenttheses.ub.rug.nl/25344/}{his bachelor thesis} to learn more about some interesting advanced topics in differential geometry.} for his comments, corrections and the appendices that he contributed to these notes.\medskip
 
-Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Jamara Admiraal, Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Anna de Bruijn, Daniel Cortlid, Harry Crane, Fionn Donogue, Jordan van Ekelenburg, Brian Elsinga, Hanneke van Harten, Martin Daan van IJcken, Mollie Jagoe Brown, Remko de Jong, Aron Karakai, Hanna Karwowska, Wietze Koops, Henrieke Krijgsheld, Justin Lin, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Alexandru Oprea, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lisa Werkman, Lars Wieringa, Federico Zadra, Tijmen van der Ree, and Jesse van der Zeijden.
+Many thanks also to the following people for their comments and for reporting a number of misprints and corrections: Jamara Admiraal, Wojtek Anyszka, Bhavya Bhikha, Huub Bouwkamp, Anna de Bruijn, Daniel Cortlid, Harry Crane, Fionn Donogue, Jordan van Ekelenburg, Brian Elsinga, Hanneke van Harten, Martin Daan van IJcken, Mollie Jagoe Brown, Nicolas Jal\`on Stewart, Remko de Jong, Aron Karakai, Hanna Karwowska, Wietze Koops, Henrieke Krijgsheld, Justin Lin, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Alexandru Oprea, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lisa Werkman, Lars Wieringa, Federico Zadra, Tijmen van der Ree, and Jesse van der Zeijden.
 
 \mainmatter