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Fix typos, add few more references
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Signed-off-by: Marcello Seri <marcello.seri@gmail.com>
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mseri committed Nov 13, 2020
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10 changes: 5 additions & 5 deletions 1-manifolds.tex
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Expand Up @@ -104,7 +104,7 @@ \section{Topological manifolds}

\begin{definition}[Topological manifold]
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
\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}}
\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}}
\begin{enumerate}[(i)]
\item $M$ is a Hausdorff space;
\item $M$ is second countable;
Expand Down Expand Up @@ -379,7 +379,7 @@ \section{Differentiable manifolds}
\end{example}

Note that smooth manifolds do not yet have a metric structure: distances between the points are not defined.
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.
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.
This allows to always view manifolds as metric spaces.

\begin{example}[A different smooth structure on $\R$]
Expand Down Expand Up @@ -684,7 +684,7 @@ \section{Smooth maps and differentiability}
\end{proposition}
\begin{proposition}
Let $M$ be a smooth manifold of dimension $n$.
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.
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.
\end{proposition}
\begin{proposition}
Let $M, N, P$ be three smooth manifolds, and suppose that $F:M\to N$ and $G:N\to P$ are smooth.
Expand Down Expand Up @@ -777,8 +777,8 @@ \section{Partitions of unity}
\end{equation}

\begin{exercise}
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 $x$.
Use this to show that $f\in C^\infty(\R)$ and that $f^{(k)}(0) = 0$ for all $k\geq 0$.
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 $x$.
Use this to show that $h\in C^\infty(\R)$ and that $h^{(k)}(0) = 0$ for all $k\geq 0$.
\end{exercise}

The function $f$ that we are seeking is then\footnote{Exercise: check that such function $f$ satisfies all the desired properties.} given by
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135 changes: 135 additions & 0 deletions aom.bbl
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Expand Up @@ -130,6 +130,46 @@
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Expand Down Expand Up @@ -195,6 +235,74 @@
\verb https://link.springer.com/book/10.1007%2F978-3-662-55774-7
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\verb{doi}
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\verb{urlraw}
\verb https://doi.org/10.1007%2F978-1-4612-0541-8
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\verb https://doi.org/10.1007%2F978-1-4612-0541-8
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Expand Down Expand Up @@ -386,6 +494,33 @@
\field{title}{Topology}
\field{year}{2000}
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\verb{urlraw}
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26 changes: 26 additions & 0 deletions aom.bib
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Expand Up @@ -60,6 +60,24 @@ @book{book:knauf
url = {https://link.springer.com/book/10.1007%2F978-3-662-55774-7}
}

@book{book:lang,
doi = {10.1007/b97450},
url = {https://doi.org/10.1007%2Fb97450},
year = 2002,
author = {Lang, Serge},
publisher = {Springer-Verlag},
title = {Introduction to Differential Manifolds}
}

@book{book:lang:infinite,
doi = {10.1007/978-1-4612-0541-8},
url = {https://doi.org/10.1007%2F978-1-4612-0541-8},
year = 1999,
publisher = {Springer New York},
author = {Serge Lang},
title = {Fundamentals of Differential Geometry}
}

@book{book:lee,
title = {Introduction to Smooth Manifolds},
author = {Lee, John M.},
Expand Down Expand Up @@ -158,3 +176,11 @@ @misc{lectures:teufel
note = {Unpublished lecture notes},
url = {http://web.archive.org/web/20201111215028/https://www.math.uni-tuebingen.de/de/forschung/maphy/personen/stefanteufel/skripte/skript2013.pdf}
}

@misc{nlab:urysohn_metrization_theorem,
author = {{nLab authors}},
title = {{{U}}rysohn metrization theorem},
url = {http://ncatlab.org/nlab/show/Urysohn%20metrization%20theorem/10},
month = nov,
year = 2020
}
5 changes: 3 additions & 2 deletions aom.tex
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Expand Up @@ -207,7 +207,7 @@
\setlength{\parskip}{\baselineskip}
Copyright \copyright\ \the\year\ \thanklessauthor

\par Version 0.4.1 -- \today
\par Version 0.4.2 -- \today

\vfill
\small{\doclicenseThis}
Expand Down Expand Up @@ -261,12 +261,13 @@ \chapter*{Introduction}
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.
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.
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.
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.

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}.
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.

I am extremely grateful to Martijn Kluitenberg for his careful reading of the notes and his useful comments and corrections.
Many thanks also to Luuk de Ridder and Huub Bouwkamp for reporting a number of misprints.
Many thanks also to Huub Bouwkamp, Luuk de Ridder and Jordan van Ekelenburg for reporting a number of misprints.

\mainmatter

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