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\title{Analysis\\ \noindent
  on\\ \noindent
  Manifolds
}
\author{Marcello Seri}
\publisher{Bernoulli Institute\\ \noindent
  %A.Y. 2022--2023\\ \noindent
  \MakeLowercase{\texttt{m.seri@rug.nl}}
}

\begin{document}
\maketitlepage

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	Copyright \copyright\ \the\year\ \thanklessauthor

	\par Version 1.6.4 -- \today

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\pagenumbering{roman}
\tableofcontents
\cleardoublepage

\pagenumbering{arabic}
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}

At the entry for \emph{Mathematical analysis}, our modern source of truth---Wikipedia---says

\begin{quotation}
	\emph{Mathematical analysis} is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions.

	These theories are usually studied in the context of real numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
\end{quotation}

\newthought{In this sense}, our course will focus on generalizing the concepts of differentiation, integration and, up to some extent, differential equations to spaces that are more general than the standard Euclidean space.
We will do this by trying to make everything look Euclidean and, in this sense, the Euclidean space $\R^n$ is going to be \emph{the} prototype of all manifolds: it won't just be our simplest example, we will see that locally every manifold looks like a Euclidean space.

Euclidean spaces, and the Riemannian charts that you may have already encountered in the Geometry course, have a very strong property: they can be described with a set of \emph{global} coordinates.
Even though this means that all computations are explicit, it does make it harder to distinguish \emph{intrinsic}\footnote{I.e. independent from the choice of coordinates.} concepts.
Manifolds will force our hand to work in a \emph{coordinate-free} setting and isolate these instrisic concepts.
We will see that this will unleash a surprising power that will allow us to lay the foundation for a lot of the mathematics that will come in the rest of the curriculum.

These notes will focus on fundamental ideas of differential geometry, in particular we will discuss manifolds, differential forms, integration, with a wink to the study of vector fields as dynamical systems and topology via cohomology.
If the time permits it, we may give a brief tour of Lie groups and Lie algebras, Riemannian metrics and the notion of curvature, or distributions and Frobenius theorem, depending on the preferences expressed in class.
Some of these topics are already present in appendices to the notes, other will be progressively added in due course.
Throughout the course and these notes, I will try to give particular emphasis on the usefulness of these topics in the mathematics of mechanics and their relevance in certain aspects of topology and field theory.

The course relies \emph{heavily} on your knowledge of linear and multilinear algebra, multivariable analysis and dynamical systems.
This should not come as a surprise: differential geometry studies the natural space in which analysis, in the sense of derivation and integration, can be performed, and was born together with classical mechanics, somehow as unique discipline, before these started diverging on their own paths.

An old mathematical joke says that
\begin{quote}
	differential geometry is the study of properties that are invariant under change of notation.
\end{quote}
Sadly, this is \emph{funny because it is alarmingly close to the truth}\footnote{Cit. Lee~\cite{book:lee}.}.

You will soon see that different references use different notations for just about everything we are going to discuss.
I'll try to stick to the ones you used in the past courses when possible, falling back to~\cite{book:lee} and~\cite{book:tu} and to my personal preference when the latter disagree.

\newthought{These lecture notes} are by no means comprehensive.
As complementery sources you can use the textbook~\cite{book:tu} or the extensive reference~\cite{book:lee}.
You should have access to both books via the University library and, in addition, Lee's ebook can be downloaded via the University proxy on \href{https://link.springer.com/book/10.1007/978-1-4419-9982-5}{SpringerLink}.

The book~\cite{book:McInerney} is a nice compact companion that develops most of the concepts of the course in the specific case of $\R^n$ and could provide further examples and food for thought.
The books \cite{book:nicolaescu}\footnote{Beware of typos, there are many.}, \cite{book:crane} and \cite{lectures:nanda}, freely available from the authors' website, are not really suitable as references for this course but provide fantastic resources for the readers that want to dig further and see where the material discussed in the course can lead.
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 (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}.\medskip

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: 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, Wietze Koops, Henrieke Krijgsheld, Valeriy Malikov, Mar\'ia Diaz Marrero, Aiva Misieviciute, Levi Moes, Nicol\'as Moro, Jard Nijholt, Magnus Petz, Jorian Pruim, Luuk de Ridder, Lisanne Sibma, Marit van Straaten, Bo Tielman, Dave Verweg, Ashwin Vishwakarma, Lars Wieringa, Federico Zadra and Jesse van der Zeijden.

\mainmatter

\chapter*{Einstein summation convention}
\addcontentsline{toc}{chapter}{Einstein summation convention}

As will become clear soon, sums of the type $\sum_i x^i e_i$ are unavoidably appearing all over the place when working on manifolds.
Therefore, throughout these notes we will apply the \emph{Einstein summation convention}: if the same index\footnote{For example, $i$ in the summation $\sum_i x^i e_i$.} appears exactly twice in a monomial term, once in the lower and once in the upper index position, then that term is understood to be summed over all possible values of that index\footnote{Usually from $1$ to the dimension of the space in question.}.

For instance, the expression
\begin{equation}
	a^{ij}b_l^k e_i e_k
\end{equation}
is a shorthand for
\begin{equation}
	\sum_{i,k} a^{ij}b_l^k e_i e_k.
\end{equation}

In general, we will use lower indices for basis of vector spaces\footnote{E.g., $(e_1,\ldots,e_n)$ could be the standard basis of $\R^n$.}, and upper indices for the components of a vector with respect to a basis\footnote{E.g., the $i$th-coordinate $x^i$ of $x\in\R^n$.}.
\marginnote[10pt]{Since the coordinates of a point $x\in\R^n$ are also its components with respect to the standard basis $(e_1, \ldots, e_n)$, for consistency they will be denoted $(x^1, \ldots, x^n)$ with upper indices.}

Note that an upper index ``in the denominator'' is regarded as a lower index, so the following are to be considered equivalent:
\begin{equation}
	\sum_{i} a^i \frac{\partial}{\partial x^i} = a^i \frac{\partial}{\partial x^i}.
\end{equation}
In fact, the expressions below are all equivalent and commonly used in the differential geometry literature:
\begin{equation}
	\sum_{i} a^i \frac{\partial}{\partial x^i} = a^i \frac{\partial}{\partial x^i} = a^i \partial_{x^i} = a^i \partial_i.
\end{equation}

\chapter{Manifolds}\label{ch:manifolds}
\input{1-manifolds}

\chapter{Tangent bundle}\label{ch:2}
\input{2-tangentbdl}

\chapter{Submanifolds}\label{ch:sub}
\input{2b-submanifolds}

\chapter{Vector fields}\label{ch:vf}
\input{3-vectorfields}

\chapter{Vector bundles}\label{sec:vectorbundle}
\input{2c-vectorbdl}

\chapter{Cotangent bundle}\label{cg:ctb}
\input{4-cotangentbdl}

\chapter{Tensor fields}\label{cg:tf}
\input{5-tensors}

\chapter{Differential forms}
\input{6-differentiaforms}

\chapter{De Rham cohomology and Poincar\'e lemma}
\input{6b-cohomology}

\chapter{Integration of forms}
\input{7-integration}

\begin{appendices}
	\chapter{Lie groups and Lie algebras}\label{appendix:Lie}
	\input{3b-liegroups}

	\input{appendices}
\end{appendices}

% \begin{appendices}
%   \chapter{Solution to selected exercises}
%   \section{Chapter~\ref{ch:manifolds}}

%   \newthought{Exercise~\ref{exe:rntopsp}.}
%   \begin{enumerate}
%     \item[] Hausdorff. For $x\neq y\in\R^n$, let $\epsilon = d(x,y)/3$.
%     Then the two balls $B_x(\epsilon) := \{z\in X \;\mid\; d(z,x)<\epsilon\}$ and $B_y(\epsilon)$ are disjoint open sets containing $x$ and $y$ respectively.
%     \item[] Second countable. As countable basis for the topology we can take the open balls $B_\epsilon(x)$ with rational radii $\epsilon\in\Q$ and centers $x\in\Q^n$.
%   \end{enumerate}

% \end{appendices}

\printbibliography
\addcontentsline{toc}{chapter}{Bibliography}
\end{document}