Further fixes from Martijn and new version

Signed-off-by: Marcello Seri <[email protected]>

3b-liegroups.tex

@@ -39,7 +39,7 @@ \section{Lie groups}
 \end{example}
 
 \begin{exercise}
-  \item Prove that any Lie group is orientable.
+  Prove that any Lie group is orientable.
 \end{exercise}
 
 \begin{definition}

6-differentiaforms.tex

@@ -413,6 +413,9 @@ \section{Exterior derivative}
      & = \sum_{i<j} \left(\frac{\partial \omega_j}{\partial x^i} - \frac{\partial \omega_i}{\partial x^j} \right) dx^i\wedge dx^j,
   \end{align}
   consistently with our previous definitions.
+
+  It is worth observing here that $d(df) = 0$ since $\frac{\partial^2 f}{\partial x^i \partial x^j} = \frac{\partial^2 f}{\partial x^j \partial x^i}$.
+  We will revisit this fact in Lemma~\ref{lem:ext_deriv}.
 \end{example}
 
 \begin{definition}
@@ -479,7 +482,7 @@ \section{Exterior derivative}
   \end{equation}
 \end{exercise}
 
-\begin{lemma}
+\begin{lemma}\label{lem:ext_deriv}
   The exterior derivative satisfies the following properties.
   For all $\omega,\omega_1,\omega_2\in\Omega^k(M)$, $\nu\in\Omega^h(M)$ and $f\in C^\infty(M)$,
   \begin{enumerate}[(i)]
@@ -548,14 +551,14 @@ \section{Exterior derivative}
   \begin{equation}
     d\eta = \left(\frac{\partial u}{\partial z}-\frac{\partial v}{\partial y} + \frac{\partial w}{\partial x}\right) dx\wedge dy \wedge dz.
   \end{equation}
-  If you compare the results we obtained above with the gradient ($\nabla$), divergence ($\nabla\cdot$) and curl ($\nabla\times$) from multivariable analysis, you would likely notice that the components of the $2$-form $\d\omega$ are exactly the components of the curl of the vector field with components $(P, Q, R)$.
+  If you compare the results we obtained above with the gradient ($\nabla$), divergence ($\nabla\cdot$) and curl ($\nabla\times$) from multivariable analysis, you would likely notice that the components of the $2$-form $d\omega$ are exactly the components of the curl of the vector field with components $(P, Q, R)$.
   Similarly, the formula for the divergence will look very close to the formula for $d\eta$.
   What is going on?
 
-  The standard euclidean metric on $\R^n$ is the metric associated to the metric tensor\footnote{Cf. Definition~\ref{def:metric}.} $g_{ij} = \delta_{ij}$.
+  The euclidean metric on $\R^n$ is the metric associated to the metric tensor\footnote{Cf. Definition~\ref{def:metric}.} $g_{ij} = \delta_{ij}$.
   We can use the musical isomorphisms\footnote{Cf. Example~\ref{ex:musicaliso}.} to identify vector fields and $1$-forms, obtaining for the components with respect to cartesian coordinates that $v_i = v^i$.
 
-  Moreover, the interior multiplication yields another map $\beta: \fX(\R^3)\to\Omega^2(\R^3)$ defined by $\beta(X) = \iota_X (dx\wedge dy\wedge dz)$, which is linear over $C^\infty(\R^3)$ (why?) and, thus, corresponds to a smooth bundle homomorphism from $T\R^3$ to $\Lambda^2\R^3$ (why?).
+  Moreover, the interior multiplication yields another map $\beta: \fX(\R^3)\to\Omega^2(\R^3)$ defined by $\beta(X) = \iota_X (dx\wedge dy\wedge dz)$, which is linear over $C^\infty(\R^3)$ (why?) and, thus, corresponds to a smooth bundle isomorphism from $T\R^3$ to $\Lambda^2\R^3$ (why?).
 
   In a similar fashion, we can also define a smooth bundle isomorphism $\bigstar: C^\infty(\R^3) \to \Omega^3(\R^3)$ via
   \begin{equation}
@@ -579,7 +582,7 @@ \section{Exterior derivative}
 
   The interest and need to generalize the operations of vector calculus in $\R^3$ to higher dimensional spaces have been one of the drives to develop the theory of differential forms.
 
-  This is a more general fact related to 3 dimensional manifolds with a Riemannian metric $M$. Then the maps defined above are still defined in the same exact way\footnote{There is a more general operator $\star : \Omega^k(M) \to \Omega^{n-k}(M)$ appearing here. This is called \emph{Hodge star} and is defined as the operator such that $\tau \wedge \star \sigma = g^\dagger(\tau,\sigma) \omega$, where $\omega := \star 1$ is the Riemannian volume and $g^\dagger$ is the inner product induced by $g$ on covector fields via the musical isomorphism between tangent and cotangent bundle. See \cite[Exercise 16-18]{book:lee} or \cite[Chapters 6.2--6.5]{book:abrahammarsdenratiu}.} up to using the correct isomorphism induced by $g$ and one obtains the following commutative diagram.
+This is a more general fact related to 3 dimensional manifolds with a Riemannian metric $M$. Then the maps defined above are still defined in the same exact way\footnote{There is a more general operator $\star : \Omega^k(M) \to \Omega^{n-k}(M)$ appearing here. This is called \emph{Hodge star} and is defined as the operator such that $\tau \wedge \star \sigma = g^\dagger(\tau,\sigma) \omega$, where $\omega := \star 1$ is the Riemannian volume and $g^\dagger$ is the inner product induced by $g$ on covector fields via the musical isomorphism between tangent and cotangent bundle (cf. Example~\ref{ex:musicaliso}). See \cite[Exercise 16-18]{book:lee} or \cite[Chapters 6.2--6.5]{book:abrahammarsdenratiu}.} up to using the correct isomorphism induced by $g$ and one obtains the following commutative diagram.
   \begin{equation}
     % https://q.uiver.app/?q=WzAsNSxbMSwxLCJcXE9tZWdhXjEoTSkiXSxbMCwwLCJcXG1hdGhjYWx7WH0oTSkiXSxbMSwyLCJcXE9tZWdhXjAoTSkiXSxbMiwyLCJcXE9tZWdhXjMoTSkiXSxbMiwxLCJcXE9tZWdhXjIoTSkiXSxbMSwwLCJcXGZsYXQiLDAseyJvZmZzZXQiOi0xfV0sWzAsMSwiXFxzaGFycCIsMCx7Im9mZnNldCI6LTF9XSxbMCw0LCJkIiwwLHsib2Zmc2V0IjotMX1dLFszLDIsIlxcc3RhciIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6ImFycm93aGVhZCJ9fX1dLFs0LDMsImQiXSxbMiwwLCJkIiwwLHsib2Zmc2V0IjoxfV0sWzAsNCwiXFxzdGFyIiwyLHsib2Zmc2V0IjoxLCJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJhcnJvd2hlYWQifX19XV0=
     \begin{tikzcd}
@@ -602,7 +605,7 @@ \section{Exterior derivative}
     \item $\mathrm{curl} := \sharp \star d\; \flat : \fX(M) \to \fX(M)$;
   \end{itemize}
   where symbols' juxtaposition means their composition.
-  Since $\sharp$ and $\flat$ are defined in terms of $g$, it becaomes clear that all those operators are tightly related to the metric.
+  Since $\sharp$ and $\flat$ are defined in terms of $g$, it becomes clear that all those operators are tightly related to the metric.
   There could be a lot more to say, but that will be more suite for a course in Riemannian geometry.
   A comprehensive free resource on the subject is \cite{book:derivations}.
 \end{example}
@@ -646,6 +649,7 @@ \section{Exterior derivative}
       \end{equation}
   \end{enumerate}
 \end{exercise}
+If you want to know more about symplectic manifolds and the role of differential geometry in classical mechanics, have a look at one of \cite{book:abrahammarsdenratiu, book:arnold, book:knauf, lectures:seri:hm}.
 
 \section{Lie derivative}
 

6b-cohomology.tex

@@ -66,11 +66,11 @@ \section{Poincar\'e lemma}
 \begin{theorem}\label{thm:deRham-invariance}
   Let $M$ be a smooth manifold and $[0,1]\times M$ the product manifold with boundary $(\{0\}\times M) \cup (\{1\}\times M) \cup ((0,1)\times\partial M)$.
   Let $i_t : M \hookrightarrow [0,1]\times M$ be the injection $i_t(p) := (t,p)$ and $\pi : [0,1]\times M \to M$ the projection onto $M$.
-  Then, there is a map
+  Then, there is a linear map
   \begin{equation}
     K : \Omega^\ell([0,1]\times M)\to \Omega^{\ell-1}(M)
   \end{equation}
-  such that for every differential $\ell$-form $\omega\in\Omega^\ell([0,1]\times M)$ one has
+  such that for every differential $\ell$-form $\omega\in\Omega^\ell([0,1]\times M)$ one has\footnote{If you took an algebraic topology course, you may recognise such map $K$ as a cochain homotopy.}
   \begin{equation}
     K(d\omega) + d(K(\omega)) = i^*_1(\omega) - i^*_0(\omega)
   \end{equation}
@@ -137,7 +137,7 @@ \section{Poincar\'e lemma}
 
   To conclude the proof, take a closed $\ell$-form on $[0,1]\times M$, then
   \begin{equation}
-    i_1^*([w]) - i_0^*([w]) = [K(d\omega) + dK(\omega)] = [dK(\omega)] =0,
+    i_1^*([\omega]) - i_0^*([\omega]) = [K(d\omega) + dK(\omega)] = [dK(\omega)] =0,
   \end{equation}
   completing the proof.
 \end{proof}
@@ -214,7 +214,9 @@ \section{Poincar\'e lemma}
   Let $F:M\to N$ and $G:N\to M$ be continuous maps such that $F\circ G$ and $G\circ F$ are homotopic to the identity maps.
   By the Whitney Approximation Theorem~\ref{thm:WhitneyApproxCont} we can approximate $F$ and $G$ by smooth maps that we keep denoting with the same symbols.
   By the previous theorem, then, $(F\circ G)^*$ and $(G\circ F)^*$ coincide with the maps induced by the identity.
-  Since $\id^*$ is clearly the identity, we see that $F^*$ is an inverse to $G^*$, which concludes the proof.
+  Since $\id^*$ is clearly the identity, we see that $F^*$ is an inverse to $G^*$.
+  Hence $F^*$ is an isomorphism between the corresponding de Rham cohomology groups,
+  concluding the proof.
 \end{proof}
 
 We are almost there.

7-integration.tex

@@ -1,4 +1,4 @@
-We finally have all the main ingredients to generalize our line integral detour and discuss integration of $n$-forms over $n$-dimensional manifolds.
+We finally have all the main ingredients to generalise our line integral detour and discuss integration of $n$-forms over $n$-dimensional manifolds.
 
 \newthought{We know from calculus one}, or our line integral examples, that the direction in which we traverse the interval, or a curve, can actually make a difference.
 As it turns out, the sign of the integral of a differential $n$-form is only fixed after choosing an orientation of the manifold.
@@ -48,7 +48,7 @@ \section{Orientation on vector spaces}
 In fact, the orientation is completely characterized by the action of $n$-forms on the bases, as the following lemma shows.
 
 \begin{lemma}\label{lemma:orient}
-  Let $V$ be a $n$-dimensional vector space and let $ \omega\in\Lambda^n(V)$ be nowhere vanishing.
+  Let $V$ be a $n$-dimensional vector space and let $\omega\in\Lambda^n(V)$ be nonzero.
   Then, all bases $\{v_1, \ldots, v_n\}$ for which $\omega(v_1,\ldots, v_n) > 0$ give the same\footnote{Not necessarily the positive orientation!} orientation for $V$.
 \end{lemma}
 % \begin{proof}
@@ -63,10 +63,10 @@ \section{Orientation on vector spaces}
   Let $V$ be a $n$-dimensional vector space, prove that two nonzero $n$-forms on $V$ determine the same orientation if and only if each is a positive multiple of the other.
 \end{exercise}
 
-This allows to define an eqivalence relation between orientations in terms of the nonzero elements in $\Lambda^n(V)$. We call these nonzero elements \emph{volume elements}.
+This allows to define an equivalence relation between orientations in terms of the nonzero elements in $\Lambda^n(V)$. We call these nonzero elements \emph{volume elements}.
 
 Two volume elements $\omega_1, \omega_2$ are equivalent if there exists $c > 0$ such that $\omega_1 = c\, \omega_2$.
-Then, the previous exercise implies that the classes of equivalence $[\omega]$ of volume elments on $V$ uniquely determine orientations on $V$.
+Then, the previous exercise implies that the classes of equivalence $[\omega]$ of volume elements on $V$ uniquely determine orientations on $V$.
 
 \begin{remark}
   Of course, if $V$ is a vector space, then an orientation on $V$ canonically determines an orientation on the dual space $V^*$ by declaring that the basis dual to a positive basis is itself positive.
@@ -76,7 +76,7 @@ \section{Orientation on vector spaces}
 
 \section{Orientation on manifolds}
 Tangent and cotangent spaces are vector spaces, so what we discussed in the previous section directly apply (as usual).
-Moreover, we can bundle up over our manifold into a vector bundle.
+Moreover, we can extend our point of view to the various tensor bundles over smooth manifold that we studied so far.
 Differential $n$-forms, then, seem a reasonable concept to define a notion of orientation for a manifold, at least if we think about their pointwise meaning of assigning an orientation to each fiber of $TM$.
 
 \begin{definition}
@@ -113,11 +113,11 @@ \section{Orientation on manifolds}
   Locally, for some chart with coordinates $(x^i)$ on $M$, we have $\omega_p = w(p)\, dx^{i_1}\wedge\cdots\wedge dx^{i_n}$ and $\eta_p = \eta(p)\, dx^{i_1}\wedge\cdots\wedge dx^{i_n}$, where both coefficients are in  $C^\infty(M)$.
   Since $\omega(p) \neq 0$ for all $p\in M$, $f(p) = \eta(p)/\omega(p)$ is a smooth function.
 
-  Conversely, if $\omega$ is a basis for $\Omega^n$, then it must be nonvanishing for all $p\in M$ since each fiber is one-dimensional.
+  Conversely, if $\omega$ is a basis\footnote{Here we mean a basis for $\Omega^n(M)$ as a $C^\infty(M)$-module, not as a vector space.} for $\Omega^n(M)$, then it must be nonvanishing for all $p\in M$ since each fiber is one-dimensional.
   \medskip
 
   \textbf{Part II: 1. $\Leftrightarrow$ 3.}
-  Assume $M$ is orientable witha volume form $\omega$.
+  Assume $M$ is orientable with a volume form $\omega$.
   By eventually restricting the domains, let the atlas $\cA = \{(U_i, \varphi_i)\}$ be such that $\varphi_i(U_i) \subset \R^n$ is connected.
   Denote $\omega_0 = dx^1 \wedge\cdots\wedge dx^n$ the standard volume form on $\R^n$, then by the previous part of the proof, $(\varphi_i)_*\omega = f_i\,\omega_0$ for some smooth function $f_i \neq 0$.
   Without loss of generality\footnote{Is it clear why?} we can assume $f_i > 0$.
@@ -164,7 +164,7 @@ \section{Orientation on manifolds}
 
 \begin{definition}
   A manifold $M$ with an oriented atlas is called \emph{oriented manifold}.
-  If an orientation exists, we call \emph{orientation} the equivalence class of atlases with the same orientation.
+  If an orientation exists, we call \emph{orientation} the equivalence class of atlases with the same orientation, i.e., the family of atlases whose union is still an oriented atlas.
   Otherwise we say that the manifold is \emph{non-orientable}.
 \end{definition}
 
@@ -175,7 +175,19 @@ \section{Orientation on manifolds}
   If $(U,\varphi)$ is a chart with local coordinates $(x^i)$ such that, in the coordinate representation, the volume form $\omega = \omega(x) dx^1\wedge\cdots\wedge dx^n$ with $\omega(x) > 0$, then we say that the chart $\varphi$ is \emph{positively oriented} with respect to $\omega$, otherwise we say that it is \emph{negatively oriented}.
   Similarly, if $M$ is connected and the charts for the oriented manifold are positively (resp. negatively) oriented, we say that the manifold is positively (resp. negatively) oriented.
 \end{definition}
-%
+
+In fact, we can go one step further and define what does it mean for a diffeomorphism to preserve or reverse orientation.
+
+\begin{definition}
+  Let $(M, \omega)$ and $(N, \eta)$ be pairs of an oriented smooth manifold and its volume form.
+  A diffeomorphism $F: M \to N$ is called \emph{orientation preserving} if $F^* \eta$ is a volume form on $M$ with the same orientation as $\omega$.
+  We call it \emph{orientation reversing} if $F^*\eta$ has the opposite orientation as $\omega$.
+\end{definition}
+
+\begin{exercise}
+  Show that a diffeomorphism $F: M \to N$ is \emph{orientation preserving} if and only if for all $p\in M$, $dF_p : T_p M \to T_{F(p)}$ is an orientation preserving as a linear map, that is, if $\det(dF_p)>0$.
+\end{exercise}
+
 % \begin{theorem}
 %   Let $M$ be a $n$-dimensional smooth manifold.
 %   A nowhere-vanishing $n$-form $\omega\in\Omega^n(M)$ uniquely determines an orientation.
@@ -227,6 +239,7 @@ \section{Orientation on manifolds}
   and $\frac{2}{1-p^2}>0$.
   If we perform the same computation on $U_2$, however, we obtain $(\varphi_2)_*(X) = -\frac{2}{1+p^2}\frac{\partial}{\partial x}\Big|_{\varphi_2(p)}$, with the negative coefficient $-\frac{2}{1+p^2} < 0$, corresponding to the opposite orientation on $U_2$.
   Of course, in this case, not all is lost: by choosing $\widetilde\varphi_2(p) = \varphi_2(-p^1, p^2)$ we obtain $(\widetilde\varphi_2)_*(X) = \frac{2}{1+p^2} \frac{\partial}{\partial x}\Big|_{\widetilde\varphi_2(p)}$ with the positive coefficient $\frac{2}{1+p^2} > 0$, which shows that $X_p$ defines an orientation on the whole $\bS^1$.
+  Note again that here the orientation is given as a positive basis for $T_p \bS^1$ at all points, not as a volume form.
 \end{example}
 
 \begin{exercise}
@@ -569,7 +582,7 @@ \section{Integrals on manifolds}
   \end{equation}
 \end{proof}
 
-This corollary has deep consequences in classical mechanics, which I am going to teach in the master and you are welcome to attend!
+This corollary has deep consequences in classical mechanics, which we teach in the master and you are welcome to attend! If you are curious you can have a look at \cite[Chapter 3]{lectures:seri:hm} and, in particular, at Liouville theorem (Theorem 3.30).
 
 \begin{remark}
   The integral defined in this section can be extended rather immediately to measurable functions.
@@ -588,7 +601,7 @@ \section{Integrals on manifolds}
 \section{Stokes' Theorem}
 
 Stokes' theorem states that if $\omega$ is an $(n-1)$-form on an orientable $n$-manifold $M$, then the integral of $d\omega$ over $M$ equals the integral of $\omega$ over $\partial M$, generalising our observations for the line integral.
-This is a beautiful and very important results, with deep consequences. The most immediate ones are the classical theorems of Gauss, Green and Stokes, which are just a special cases of this result.
+This is a beautiful and very important result, with deep consequences. The most immediate ones are the classical theorems of Gauss, Green and Stokes, which are just a special cases of this result.
 
 We are going to state the theorem, discuss some of its consequences and then give its proof.
 

aom.bib

@@ -208,7 +208,15 @@ @misc{lectures:merry
   title  = {Lecture notes on Differential Geometry},
   year   = 2019,
   note   = {Unpublished lecture notes},
-  url    = {https://www.merry.io/courses/differential-geometry/}
+  url    = {https://web.archive.org/web/20231106145652/https://www2.math.ethz.ch/will-merry/files/Merry%20-%20Differential%20Geometry%20%282019%29.pdf}
+}
+
+@misc{lectures:seri:hm,
+  author = {Seri, Marcello},
+  title  = {Hamiltonian Mechanics},
+  year   = {2023},
+  note   = {Unpublished lecture notes, version 1.11},
+  url    = {https://github.com/mseri/hammech20/releases/tag/1.11}
 }
 
 @misc{lectures:teufel,

aom.tex

@@ -214,7 +214,7 @@
   \setlength{\parskip}{\baselineskip}
   Copyright \copyright\ \the\year\ \thanklessauthor
 
-  \par Version 1.5 -- \today
+  \par Version 1.6 -- \today
 
   \vfill
   \small{\doclicenseThis}
@@ -270,7 +270,7 @@ \chapter*{Introduction}
 Finally, a colleague mentioned~\cite{book:lang}. I don't have experience with this book but from a brief look it seems to follow a similar path as these lecture notes, so it might provide yet an alternative reference after all.
 
 The idea for the cut that I want to give to this course was inspired by the online \href{https://www.video.uni-erlangen.de/course/id/242}{Lectures on the Geometric Anatomy of Theoretical Physics} by Frederic Schuller, by the lecture notes of Stefan Teufel's Classical Mechanics course~\cite{lectures:teufel} (in German), by the classical mechanics book by Arnold~\cite{book:arnold} and by the Analysis of Manifolds chapter in~\cite{book:thirring}.
-In some sense I would like this course to provide the introduction to geometric analysis that I wish was there when I prepared my \href{https://www.mseri.me/lecture-notes-hamiltonian-mechanics/}{first edition} of the Hamiltonian mechanics course.
+In some sense I would like this course to provide the introduction to geometric analysis that I wish was there when I prepared my \href{https://www.mseri.me/lecture-notes-hamiltonian-mechanics/}{first edition} of the Hamiltonian mechanics course (see also my lecture notes for that course \cite{lectures:seri:hm}).
 In addition to the reference above, these lecture notes have found deep inspiration from~\cite{lectures:merry,lectures:hitchin} (all freely downloadable from the respective authors' websites), and from the book~\cite{book:abrahammarsdenratiu}.
 
 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.