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Fix typos, add few more references
Signed-off-by: Marcello Seri <[email protected]>
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‎1-manifolds.tex‎

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@@ -104,7 +104,7 @@ \section{Topological manifolds}
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\begin{definition}[Topological manifold]
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A topological space\footnote{From now on, if we say that $X$ is a topological space we are implying that there is a topology $\cT$ defined on $X$.} $M$ is a \emph{topological manifold} of dimension $n$, or topological $n$-manifold, if it has the following properties
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\marginnote[0.5em]{Note that the finite dimensionality is a somewhat artificial restriction: manifolds can be infinitely dimensional. For example, the space of continuous functions between manifolds is a so-called infinite-dimensional Banach manifold.\vspace{1em}}
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\marginnote[0.5em]{Note that the finite dimensionality is a somewhat artificial restriction: manifolds can be infinitely dimensional~\cite{book:lang:infinite}. For example, the space of continuous functions between manifolds is a so-called infinite-dimensional Banach manifold.\vspace{1em}}
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\begin{enumerate}[(i)]
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\item $M$ is a Hausdorff space;
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\item $M$ is second countable;
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\end{example}
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Note that smooth manifolds do not yet have a metric structure: distances between the points are not defined.
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However, they are \emph{metrizable}\footnote{In fact, all the topological manifolds are metrizable. This property is far more general and harder to prove~\cite[Theorem 34.1 and Exercise 1 of Chapter 4.36]{book:munkres:topology}.}: there exists some metric on the manifold that induces the given topology on it.
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However, they are \emph{metrizable}\footnote{In fact, all the topological manifolds are metrizable. This property is far more general and harder to prove~\cite[Theorem 34.1 and Exercise 1 of Chapter 4.36]{book:munkres:topology} or \cite{nlab:urysohn_metrization_theorem}. Note that not all topological spaces are metrizable, for example a space with more than one point endowed with the discrete topology is not. And even if a topological space is metrizable, the metric will be far from unique: for example, proportional metrics generate the same collection of open sets.}: there exists some metric on the manifold that induces the given topology on it.
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This allows to always view manifolds as metric spaces.
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\begin{example}[A different smooth structure on $\R$]
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\end{proposition}
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\begin{proposition}
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Let $M$ be a smooth manifold of dimension $n$.
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Then $f:\R^m\to M$ is smooth iff for all charts $(U,\varphi)$ of $M$, the function $\varphi\circ F:F^{-1}(U)\to\R^m$ is smooth.
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Then $F:\R^m\to M$ is smooth iff for all charts $(U,\varphi)$ of $M$, the function $\varphi\circ F:F^{-1}(U)\to\R^m$ is smooth.
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\end{proposition}
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\begin{proposition}
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Let $M, N, P$ be three smooth manifolds, and suppose that $F:M\to N$ and $G:N\to P$ are smooth.
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\end{equation}
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\begin{exercise}
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Prove by induction that for $t>0$ and $k\geq 0$, the $k$th derivative $f^{(k)}(t)$ is of the form $p_{2k}(1/t)e^{-1/t}$ for some polynomial $p_{2k}(x)$ of degree $2k$ in $x$.
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Use this to show that $f\in C^\infty(\R)$ and that $f^{(k)}(0) = 0$ for all $k\geq 0$.
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Prove by induction that for $t>0$ and $k\geq 0$, the $k$th derivative $h^{(k)}(t)$ is of the form $p_{2k}(1/t)e^{-1/t}$ for some polynomial $p_{2k}(x)$ of degree $2k$ in $x$.
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Use this to show that $h\in C^\infty(\R)$ and that $h^{(k)}(0) = 0$ for all $k\geq 0$.
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\end{exercise}
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The function $f$ that we are seeking is then\footnote{Exercise: check that such function $f$ satisfies all the desired properties.} given by

‎aom.bbl‎

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\verb https://link.springer.com/book/10.1007%2F978-3-642-22597-0
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\endverb
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\endentry
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\field{labeltitlesource}{title}
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\field{month}{3}
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\field{title}{Differential Forms}
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\field{year}{2019}
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\verb{doi}
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\verb 10.1142/11058
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\endverb
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\verb{urlraw}
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\verb http://web.archive.org/web/20200525224100/https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf
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\endverb
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\verb{url}
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\verb http://web.archive.org/web/20200525224100/https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf
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\endverb
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\endentry
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\entry{lectures:hitchin}{misc}{}
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\name{author}{1}{}{%
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{{hash=d0af63589d0e61aaeda653b8860bf668}{%
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\verb https://link.springer.com/book/10.1007%2F978-3-662-55774-7
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\endverb
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\endentry
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\field{labelnamesource}{author}
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\field{labeltitlesource}{title}
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\field{title}{Introduction to Differential Manifolds}
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\field{year}{2002}
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\verb{doi}
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\verb 10.1007/b97450
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\endverb
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\verb{urlraw}
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\verb https://doi.org/10.1007%2Fb97450
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\endverb
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\verb{url}
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\verb https://doi.org/10.1007%2Fb97450
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\endverb
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{Springer New York}%
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\field{title}{Fundamentals of Differential Geometry}
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\field{year}{1999}
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\verb{doi}
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\verb 10.1007/978-1-4612-0541-8
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\endverb
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\verb{urlraw}
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\verb https://doi.org/10.1007%2F978-1-4612-0541-8
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\endverb
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\verb{url}
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\verb https://doi.org/10.1007%2F978-1-4612-0541-8
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\endverb
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\endentry
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\entry{book:lee:topology}{book}{}
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\name{author}{1}{}{%
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{{hash=32fbdee07151a2283306531745546911}{%
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\field{title}{Topology}
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\field{year}{2000}
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\endentry
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\entry{nlab:urysohn_metrization_theorem}{misc}{}
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\field{month}{11}
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\field{title}{{{U}}rysohn metrization theorem}
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\field{year}{2020}
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\verb{urlraw}
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\verb http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10
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\endverb
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\verb{url}
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\verb http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10
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\endverb
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\endentry
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\entry{lectures:teufel}{misc}{}
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\name{author}{1}{}{%
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{{hash=91396cef9166c4f47cfbe40ba6179b13}{%

‎aom.bib‎

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url = {https://link.springer.com/book/10.1007%2F978-3-662-55774-7}
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}
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@book{book:lang,
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doi = {10.1007/b97450},
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url = {https://doi.org/10.1007%2Fb97450},
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year = 2002,
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author = {Lang, Serge},
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publisher = {Springer-Verlag},
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title = {Introduction to Differential Manifolds}
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}
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@book{book:lang:infinite,
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doi = {10.1007/978-1-4612-0541-8},
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url = {https://doi.org/10.1007%2F978-1-4612-0541-8},
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year = 1999,
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publisher = {Springer New York},
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author = {Serge Lang},
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title = {Fundamentals of Differential Geometry}
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}
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@book{book:lee,
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title = {Introduction to Smooth Manifolds},
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author = {Lee, John M.},
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note = {Unpublished lecture notes},
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url = {http://web.archive.org/web/20201111215028/https://www.math.uni-tuebingen.de/de/forschung/maphy/personen/stefanteufel/skripte/skript2013.pdf}
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}
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@misc{nlab:urysohn_metrization_theorem,
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author = {{nLab authors}},
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title = {{{U}}rysohn metrization theorem},
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url = {http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10},
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month = nov,
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year = 2020
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}

‎aom.tex‎

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\setlength{\parskip}{\baselineskip}
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Copyright \copyright\ \the\year\ \thanklessauthor
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\par Version 0.4.1 -- \today
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\par Version 0.4.2 -- \today
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\vfill
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\small{\doclicenseThis}
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I have requested for~\cite{book:tu} book to be freely available via SpringerLink using the university proxy but this will take some time to become active.
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However, you can already freely access Lee's book via the University proxy on \href{https://link.springer.com/book/10.1007/978-1-4419-9982-5}{SpringerLink} and it will provide a very good and extensive reference for this and other future courses.
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The book~\cite{book:McInerney} is a nice compact companion that develops most of this course concept in the specific case of $\R^n$ and could provide further examples and food for thoughts.
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A colleague recently mentioned also~\cite{book:lang}. I don't know this book but from a brief look it seems to follow a similar path as these lecture notes, so might provide an alternative reference after all.
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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 Manifold chapter in~\cite{book:thirring}.
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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.
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I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his useful comments and corrections.
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Many thanks also to Luuk de Ridder and Huub Bouwkamp for reporting a number of misprints.
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Many thanks also to Huub Bouwkamp, Luuk de Ridder and Jordan van Ekelenburg for reporting a number of misprints.
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\mainmatter
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