From 76a703731edf752c847d1a7527aaac1dc7bd8d03 Mon Sep 17 00:00:00 2001 From: Marcello Seri Date: Wed, 10 Feb 2021 09:52:44 +0100 Subject: [PATCH] Big update to cleanup some more todos and typos Signed-off-by: Marcello Seri --- 1-manifolds.tex | 76 +++++++++++++++++++++++----------------------- 2-tangentbdl.tex | 50 ++++++++++++++++-------------- 3-vectorfields.tex | 2 +- 3b-liegroups.tex | 28 +++++++++++++++-- 4-cotangentbdl.tex | 24 +++++++++------ 5-tensors.tex | 2 +- 7-integration.tex | 30 +++++++++++++++++- TODO.md | 8 ----- aom.tex | 2 +- 9 files changed, 136 insertions(+), 86 deletions(-) diff --git a/1-manifolds.tex b/1-manifolds.tex index b90a1e6..453ca94 100644 --- a/1-manifolds.tex +++ b/1-manifolds.tex @@ -285,9 +285,9 @@ \section{Differentiable manifolds} Why is this independent of the choice of the isomorphism $T$? This fact has a very interesting consequence. - The space $\mathrm{Mat}(n)$ of $n\times n$-matrices can be identified with $\R^{n^2}$ by writing the elements of the matrix as a $n^2$-vector. - This gives to $\mathrm{Mat}(n)$ a structure of differentiable manifold. - The subset of invertible matrices $GL(n) = \{ A \in \mathrm{Mat}(n) \;\mid\; \det A \neq 0\}$, widely known as the \emph{general linear group}, being an open subset of $\mathrm{Mat}(n)$ (why?) is itself a differentiable manifold. + The space $\mathrm{Mat}(n, \R)$ of real $n\times n$-matrices can be identified with $\R^{n^2}$ by writing the elements of the matrix as a $n^2$-vector. + This gives to $\mathrm{Mat}(n, \R)$ a structure of differentiable manifold. + The subset of invertible matrices $GL(n) := GL(n, \R) = \{ A \in \mathrm{Mat}(n, \R) \;\mid\; \det A \neq 0\}$, widely known as the \emph{general linear group}, being an open subset of $\mathrm{Mat}(n, \R)$ (why?) is itself a differentiable manifold. Is such manifold connected? Why? \end{exercise} @@ -382,41 +382,6 @@ \section{Differentiable manifolds} 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 $\R$] - Consider the homeomorphism $\psi:\R\to\R$, $\psi(x) = x^3$. - The atlas consisting of the global chart $(\R, \psi)$ defines a smooth structure on $\R$. - This chart is not smoothly compatible with the standard smooth structure on $\R$ since $\id_\R \circ \psi^{-1} (y) = y^{1/3}$ is not smooth at $y=0$. - Therefore, the smooth structure defined on $\R$ by $\psi$ is different from the standard one. - You can adapt this idea to construct many different smooth structures on topological manifolds provided that they at least have one smooth structure. -\end{example} - -\begin{exercise} - Show that there exists a diffeomorphism between the smooth structures $(\R, \id_\R)$ and $(\R, \psi)$ from the previous example. -\end{exercise} - -\begin{exercise}[\textit{[homework 1]}] - For $r>0$, let $\phi_r:\R\to\R$ be the map given by - \begin{equation} - \phi_r(t) := \begin{cases} - t, & \mbox{if } t<0,\\ - rt, & \mbox{if } t\geq0. - \end{cases} - \end{equation} - Let $\cA_r$ denote the maximal atlas on $\R$ containing the chart $(\R, \phi_r)$. - \begin{enumerate} - \item Show that the differentiable structures on $\R$ defined by $\cA_r$ and $\cA_s$, $00$. - \end{enumerate} -\end{exercise} - -\begin{remark} - There exist examples of topological manifolds without smooth structures. - It is also known that smooth manifolds of dimension $n < 4$ have exactly one smooth structure (up to diffeomorphisms) while ones of dimension $n > 4$ have finitely many\footnote{A beautiful example of this is the $7$-sphere $\bS^7$ which is known to have 28 non-diffeomorphic smooth structures.}. - The case $n = 4$ is unknown: if you prove that there is only one smooth structure, you will have shown the smooth Poincar\'e conjecture. -\end{remark} - Instead of always constructing a topological manifold and then specify a smooth structure, it is often convenient to combine these steps into a single construction. This is especially useful when the initial set is not equipped with a topology. In this respect, the following lemma provides a welcome shortcut: in brief it says that given a set with suitable ``charts'' that overlap smoothly, we can use those to define both a topology and a smooth structure on the set. @@ -722,6 +687,41 @@ \section{Smooth maps and differentiability} From now on, when we write manifold, chart, atlas, etc. we always mean smooth manifold, smooth chart, smooth atlas, etc.. \end{tcolorbox} +\begin{example}[A different smooth structure on $\R$] + Consider the homeomorphism $\psi:\R\to\R$, $\psi(x) = x^3$. + The atlas consisting of the global chart $(\R, \psi)$ defines a smooth structure on $\R$. + This chart is not smoothly compatible with the standard smooth structure on $\R$ since $\id_\R \circ \psi^{-1} (y) = y^{1/3}$ is not smooth at $y=0$. + Therefore, the smooth structure defined on $\R$ by $\psi$ is different from the standard one. + You can adapt this idea to construct many different smooth structures on topological manifolds provided that they at least have one smooth structure. +\end{example} + +\begin{exercise} + Show that there exists a diffeomorphism between the smooth structures $(\R, \id_\R)$ and $(\R, \psi)$ from the previous example. +\end{exercise} + +\begin{exercise}[\textit{[homework 1]}] + For $r>0$, let $\phi_r:\R\to\R$ be the map given by + \begin{equation} + \phi_r(t) := \begin{cases} + t, & \mbox{if } t<0,\\ + rt, & \mbox{if } t\geq0. + \end{cases} + \end{equation} + Let $\cA_r$ denote the maximal atlas on $\R$ containing the chart $(\R, \phi_r)$. + \begin{enumerate} + \item Show that the differentiable structures on $\R$ defined by $\cA_r$ and $\cA_s$, $00$. + \end{enumerate} +\end{exercise} + +\begin{remark} + There exist examples of topological manifolds without smooth structures. + It is also known that smooth manifolds of dimension $n < 4$ have exactly one smooth structure (up to diffeomorphisms) while ones of dimension $n > 4$ have finitely many\footnote{A beautiful example of this is the $7$-sphere $\bS^7$ which is known to have 28 non-diffeomorphic smooth structures.}. + The case $n = 4$ is unknown: if you prove that there is only one smooth structure, you will have shown the smooth Poincar\'e conjecture. +\end{remark} + \section{Partitions of unity} \newthought{Cutoff functions} are a class of smooth functions that will be of crucial importance throughout the course, and whose existence cannot be given for granted. diff --git a/2-tangentbdl.tex b/2-tangentbdl.tex index 7f41fc8..739255a 100644 --- a/2-tangentbdl.tex +++ b/2-tangentbdl.tex @@ -596,7 +596,7 @@ \section{The differential of a smooth map}\label{sec:diffsmooth} An important consequence of what we have seen so far is that we can routinely \emph{identify} tangent vectors to a finite-dimensional vector space with elements of the space itself. More generally, if $M$ is an open submanifold of a vector space $V$, we can combine the identifications $T_p M \simeq T_p V \simeq V$ to obtain a canonical identification of each tangent space to $M$ with $V$. -For example, since $GL_n(\R)$ is an open submanifold of the vector space $\mathrm{Mat}(n)$, we can identify its tangent space at each point $X\in GL_n(\R)$ with the full space of matrices $\mathrm{Mat}(n)$. +For example, since $GL_n(\R)$ is an open submanifold of the vector space $\mathrm{Mat}(n, \R)$, we can identify its tangent space at each point $X\in GL_n(\R)$ with the full space of matrices $\mathrm{Mat}(n, \R)$. \begin{exercise}[Tangent space of a product manifold] Let $M_1, \ldots, M_k$ be smooth manifolds (without boundary\sidenote[][-8em]{The statement is true also if one (only one!) of the $M_i$ spaces is a smooth manifold with boundary. If there is more than one manifold with boundary, the product space will have ``corners'' that cannot be mapped to half spaces and thus is not a smooth manifold, as a simple example you can consider the closed square $[0,1]\times [0,1]$.}), and for each $j$ let $\pi_j:M_1\times\cdots\times M_k \to M_j$ be the projection onto the $M_j$ factor. @@ -1218,24 +1218,6 @@ \section{Submanifolds} \end{enumerate} \end{exercise} -\newthought{Of course, we can also define subbundles}. -\begin{definition} - Let $\pi:E \to M$ be a rank-$n$ vector bundle and $F\subset E$ a submanifold. - If for all $p\in M$, the intersection $F_p := F\cap E_p$ is a $k$-dimensional subspace of the vector space $E_p$ and $\pi|_F : F \to M$ defines a rank-$k$ vector bundle, then $\pi|_F: F \to M$ is called a \emph{subbundle} of $E$. -\end{definition} - -\begin{exercise}[\textit{[homework 2]}] - Let $M$ be a smooth $m$-manifold and $N$ a smooth $n$-manifold. - Let $F:M\to N$ be an embedding and denote $\widetilde M = F(M)\subset N$. - \begin{enumerate} - \item Show that the tangent bundle of $M$ in $N$, given by $T\widetilde M := dF(TM) \subset TN\big|_{\widetilde M}$, is a subbundle of $TN\big|_{\widetilde M}$ by providing explicit local trivialisations in terms of the charts $(U, \varphi)$ for $M$. - \item Assume that there exist a smooth function $\Phi:N\to\R^{n-m}$ such that $\widetilde M := \{p\in N \mid \Phi(p) = 0\}$ and $d\Phi_p$ has full rank for all $p\in\widetilde M$. Prove that - \begin{equation} - T\widetilde{M} = \{(p,v)\in TN|_{\widetilde{M}} \mid v\in\ker(d\Phi_p)\}. - \end{equation} - \end{enumerate} -\end{exercise} - \newthought{We still have a question pending} since the beginning of the chapter. Is the tangent space to a sphere the one that we naively imagine (see Figure~\ref{fig:tan-embedded-sphere})? To finally answer the question, we will prove one last proposition. @@ -1287,16 +1269,38 @@ \section{Submanifolds} \end{exercise} \begin{exercise}[\textit{[homework 2]}]\label{exe:onsubmanifold} - Show that the orthogonal matrices $O(n) := \{ Q\in GL(n) \mid Q^TQ=\id \}$ form a $n(n-1)/2$-dimensional submanifold of the $n^2$-manifold $\mathrm{Mat}(n)$ of $n\times n$-matrices. + Show that the orthogonal matrices + \begin{equation} + O(n) := O(n, \R) = \{ Q\in \mathrm{Mat}(n, \R) \mid Q^TQ=\id \} + \end{equation} + form a $n(n-1)/2$-dimensional submanifold of the $n^2$-manifold $\mathrm{Mat}(n, \R)$ of $n\times n$-matrices. Show also that \begin{equation} - T_Q O(n) = \left\lbrace B \in \mathrm{Mat}(n) \mid (Q^{-1} B)^T = -Q^{-1}B \right\rbrace, + T_Q O(n) = \left\lbrace B \in \mathrm{Mat}(n, \R) \mid (Q^{-1} B)^T = -Q^{-1}B \right\rbrace, \end{equation} and, thus, that $T_{\id} O(n)$ is the space of skew-symmetric matrices \begin{equation} - T_{\id} O(n) = \left\{ B \in \mathrm{Mat}(n) \mid B^T = -B \right\}. + T_{\id} O(n) = \left\{ B \in \mathrm{Mat}(n, \R) \mid B^T = -B \right\}. \end{equation} - \textit{\small Hint: Find a suitable map $F: \mathrm{Mat}(n) \to \mathrm{Sym}(n)$ such that $F^{-1}(\{p\}) = O(n)$ for some point $p$ in the image, e.g. $0$ or $\id_n$. + \textit{\small Hint: Find a suitable map $F: \mathrm{Mat}(n, \R) \to \mathrm{Sym}(n)$ such that $F^{-1}(\{p\}) = O(n)$ for some point $p$ in the image, e.g. $0$ or $\id_n$. Here $\mathrm{Sym}(n)$ denotes the space of symmetric matrices.} \end{exercise} + +\newthought{Of course, we can also define subbundles}. +\begin{definition} + Let $\pi:E \to M$ be a rank-$n$ vector bundle and $F\subset E$ a submanifold. + If for all $p\in M$, the intersection $F_p := F\cap E_p$ is a $k$-dimensional subspace of the vector space $E_p$ and $\pi|_F : F \to M$ defines a rank-$k$ vector bundle, then $\pi|_F: F \to M$ is called a \emph{subbundle} of $E$. +\end{definition} + +\begin{exercise}[\textit{[homework 2]}] + Let $M$ be a smooth $m$-manifold and $N$ a smooth $n$-manifold. + Let $F:M\to N$ be an embedding and denote $\widetilde M = F(M)\subset N$. + \begin{enumerate} + \item Show that the tangent bundle of $M$ in $N$, given by $T\widetilde M := dF(TM) \subset TN\big|_{\widetilde M}$, is a subbundle of $TN\big|_{\widetilde M}$ by providing explicit local trivialisations in terms of the charts $(U, \varphi)$ for $M$. + \item Assume that there exist a smooth function $\Phi:N\to\R^{n-m}$ such that $\widetilde M := \{p\in N \mid \Phi(p) = 0\}$ and $d\Phi_p$ has full rank for all $p\in\widetilde M$. Prove that + \begin{equation} + T\widetilde{M} = \{(p,v)\in TN|_{\widetilde{M}} \mid v\in\ker(d\Phi_p)\}. + \end{equation} + \end{enumerate} +\end{exercise} diff --git a/3-vectorfields.tex b/3-vectorfields.tex index eba5eb0..4caf66f 100644 --- a/3-vectorfields.tex +++ b/3-vectorfields.tex @@ -344,7 +344,7 @@ \section{Lie brackets} This may seem an alien concept at first, however there are many simple examples of Lie algebras. To name a few: \begin{enumerate} \item $\R^3$ with the cross product $[x,y]:=x\times y$ is a $3$-dimensional Lie algebra; - \item the set $\mathrm{Mat}(n)$ of $n\times n$-matrices with the matrix commutator $[A,B] = AB-BA$ is a $n^2$-dimensional Lie algebra, usually denoted $\mathfrak{gl}(n, \R)$; + \item the set $\mathrm{Mat}(n, \R)$ of $n\times n$-matrices with the matrix commutator $[A,B] = AB-BA$ is a $n^2$-dimensional Lie algebra, usually denoted $\mathfrak{gl}(n, \R)$; \item any vector space $V$ turns into an (abelian) Lie algebra by defining $[v,w]=0$; \item if $V$ is a vector space, the vector space of all linear maps from $V$ to itself becomes a Lie algebra, denoted $\mathfrak{gl}(V)$, with the brackets defined by $[A,B] = A\circ B-B\circ A$. Note that with the usual identification of $n\times n$ matrices with linear maps from $\R^n$ to itself, $\mathfrak{gl}(\R^n)$ coincides with $\mathfrak{gl}(n, \R)$. \end{enumerate} diff --git a/3b-liegroups.tex b/3b-liegroups.tex index 00ef837..66c06b6 100644 --- a/3b-liegroups.tex +++ b/3b-liegroups.tex @@ -133,6 +133,27 @@ \section{Lie groups} You have proven this when you solved Exercise~\ref{exe:onsubmanifold}. \end{example} +\begin{exercise}\label{ex:SL2LGA} + For this exercise is useful to remember that we can identify the space $\mathrm{Mat}(2,\R)$ of $2\times 2$-matrices with $\R^4$ by associating the matrix $X = \begin{pmatrix}x_{11} & x_{12}\\ x_{21} & x_{22}\end{pmatrix}$ with the point $(x_{11}, x_{12}, x_{21}, x_{22})\in\R^4$. + + \begin{enumerate} + \item Show that the set + \begin{equation*} + \mathrm{SL}(2) := \mathrm{SL}(2,\R) = \{ A\in \mathrm{Mat}(2,\R) \;\mid\; \det A = 1 \} + \end{equation*} + is a 3-dimensional smooth submanifold of $\mathrm{Mat}(2,\R)$. + \item Let $e\in\mathrm{Mat}(2,\R)$ denote the identity matrix. Show that + \begin{equation*} + T_e \mathrm{SL}(2) = \{ A\in \mathrm{Mat}(2,\R) \;\mid\; \mathrm{tr}\, A = 0 \}, + \end{equation*} + where $\mathrm{tr} A$ denotes the matrix trace, i.e., the sum of the diagonal entries of $A$. + \item Let $\iota: \mathrm{SL}(2)\to \mathrm{SL}(2)$ be the map $\iota(A) = A^{-1}$. Show that $\iota$ is smooth. + \item Show that $d\iota_e: T_e\mathrm{SL}(2)\to T_e\mathrm{SL}(2)$ is given by $d\iota_e(A) = -A$. + \end{enumerate} +\end{exercise} + +In fact, some parts of the Exercise~\ref{ex:SL2LGA} above are instances of a them more general statement of Exercise~\ref{ex:DiffGroupMaps}. + \begin{exercise}\label{ex:DiffGroupMaps} Let $G$ be a Lie group. \begin{enumerate} @@ -163,13 +184,14 @@ \section{Lie algebras} \begin{example} \begin{enumerate} - \item The Lie algebra of $GL(n)$ is $\mathfrak{gl}(n)\simeq \mathrm{Mat}(n)$. + \item The Lie algebra of $GL(n)$ is $\mathfrak{gl}(n)\simeq \mathrm{Mat}(n, \R)$. \item The Lie algebra of $O(n)$ is $\mathfrak{o}(n) = \{A \in \mathfrak{gl}(n) \mid A+A^T = 0\}$. You have shown it in Exercise~\ref{exe:onsubmanifold}. + \item Can you guess what is the Lie algebra of $SL(2)$ from Exercise~\ref{ex:SL2LGA}? \end{enumerate} \end{example} \begin{exercise} - The Lie algebra of $\bT^n$ is $\R^n$.\\ + Show that the Lie algebra of $\bT^n$ is $\R^n$.\\ \textit{\small Hint: using the fact that $T(M\times N) \simeq T(M)\times T(N)$ and look at what happens in the case $n=1$.} \end{exercise} @@ -327,7 +349,7 @@ \section{Lie algebras} Use the inclusion $i:H\hookrightarrow G$ as the homomorphism, then $di_e:\fh = T_eH\to \fg = T_eG$ is the Lie algebra homomorphism. \end{proof} -If we go back to the example of $GL(n)$, now we have two possibly different Lie brackets on $\mathfrak{gl}(n)=\mathrm{Mat}(n)$: the one coming from the previous corollary and the matrix commutator. +If we go back to the example of $GL(n)$, now we have two possibly different Lie brackets on $\mathfrak{gl}(n)=\mathrm{Mat}(n, \R)$: the one coming from the previous corollary and the matrix commutator. The next result, which we will not prove, shows that they coincide. \marginnote{See~\cite[Proposition 8.41]{book:lee} for reference.} diff --git a/4-cotangentbdl.tex b/4-cotangentbdl.tex index 1947c35..e067cdd 100644 --- a/4-cotangentbdl.tex +++ b/4-cotangentbdl.tex @@ -245,11 +245,11 @@ \section{One-forms and the cotangent bundle} The cotangent bundle is a vector bundle of rank $n$ with projection $\pi:T^*M\to M$, $(p,\omega)\mapsto p$. The cotangent spaces are the fibres of the cotangent bundle. -\begin{theorem} +\begin{theorem}\label{thm:starmbld} Let $M$ be a smooth $n$-manifold. - The smooth structure on $M$ naturally induces a smooth structure on $T^*M$, making $T^*M$ into a smooth manifold of dimension $2n$ for which all coordinate covector fields are smooth local sections. + The smooth structure on $M$ naturally induces a smooth structure on $T^*M$, making $T^*M$ into a smooth manifold of dimension $2n$. \end{theorem} -\begin{proof} +%\begin{proof} % The proof is analogous to that of Theorem~\ref{thm:tgbdlsmoothmfld}, so we will not do it again. % The atlas is obtained by an atlas $\{(U_i, \varphi_i)\}$ of $M$ by defining the new atlas % \begin{equation} @@ -260,21 +260,25 @@ \section{One-forms and the cotangent bundle} % \left(\varphi_i, (\varphi_i^{-1})^*\right) : T^* U_i &\to T^*\varphi(U_i),\\ % (p, \omega) &\mapsto \left(\varphi_i(p), d(\varphi_i^{-1})^*)_p\omega \right). % \end{align} - \begin{exercise}\textit{[homework 3]} - Mimicking what we did for Theorem~\ref{thm:tgbdlsmoothmfld}, complete this proof. - \end{exercise} -\end{proof} +\begin{exercise}\label{exe:prooftstarmbld}\textit{[homework 3]} + Mimicking what we did for Theorem~\ref{thm:tgbdlsmoothmfld}, complete the proof of Theorem~\ref{thm:starmbld}. +\end{exercise} +%\end{proof} -\begin{definition} +\begin{definition}\label{def:covfield} A \emph{covector field} or a \emph{(differential) $1$-form} on $M$ is a smooth section of $T^*M$. That is, a $1$-form $\omega\in\Gamma(T^*M)$ is a smooth map $\omega: p \to \omega_p \in T_p^*M$ that assigns to each point $p\in M$ a cotangent vector at $p$. We denote the space of all smooth covector fields on $M$ by $\fX^*(M)$. \marginnote[-1em]{For reasons related to tensor fields that we will understand soon, this is sometimes denoted $\cT_1^0(M)$.} - As for vector fields, we can define \emph{$C^p$-covector fields} as the $C^p$-maps $\omega:M\to T^*M$ such that $\pi\circ\omega = \id_M$. + Of course, one can also define \emph{$C^p$-covector fields} as the $C^p$-maps $\omega:M\to T^*M$ such that $\pi\circ\omega = \id_M$. \end{definition} -Also in this case, we will often identify for a covector field $\omega\in\fX^*(M)$ its value $\omega(p) = \omega_p \in \{p\}\times T^*_p M$ at $p\in M$ with its part in $T_p^*M$ without necessarily making this explicit in the notation by projecting on the second factor. +\begin{remark} + In Exercise~\ref{exe:prooftstarmbld} you have shown that coordinate covector fields are smooth local sections for the cotangent bundle. +\end{remark} + +In analogy with the previous chapters, for covector fields $\omega\in\fX^*(M)$ we will often make the identification of its value $\omega(p) = \omega_p \in \{p\}\times T^*_p M$ at $p\in M$ with its part in $T_p^*M$ without necessarily making this explicit in the notation by projecting on the second factor. \begin{example} Let $f\in C^\infty(M)$, then the map diff --git a/5-tensors.tex b/5-tensors.tex index 52524d1..71d9b4c 100644 --- a/5-tensors.tex +++ b/5-tensors.tex @@ -8,7 +8,7 @@ \begin{itemize} \item a scalar product is a bilinear map $\langle\cdot,\cdot\rangle:V\times V\to \R$; \item the signed area spanned by two vectors is a bilinear map $\R^2\times\R^2\to\R$ defined by $\mathrm{area}(u,v) := u\wedge v = u^1v^2-u^2v^1$; - \item the determinant\footnote{In fact, the signed area is the determinant of the $2\times 2$ matrix $(u \, v)$...} of a square matrix in $\mathrm{Mat}(n)$, viewed as a function $\det: \LaTeXunderbrace{\R^n\times\cdots\times\R^n}_{n\mbox{ times}}\to\R$ is a $n$-linear map. + \item the determinant\footnote{In fact, the signed area is the determinant of the $2\times 2$ matrix $(u \, v)$...} of a square matrix in $\mathrm{Mat}(n, \R)$, viewed as a function $\det: \LaTeXunderbrace{\R^n\times\cdots\times\R^n}_{n\mbox{ times}}\to\R$ is a $n$-linear map. \end{itemize} So functions of several vectors or covectors that are linear in each argument are also called multilinear forms or tensors. diff --git a/7-integration.tex b/7-integration.tex index f837508..d898524 100644 --- a/7-integration.tex +++ b/7-integration.tex @@ -451,7 +451,7 @@ \section{Integrals on manifolds} In this exercise we are going to define a canonical volume $\omega_n$ on $\bS^n$ and prove that \begin{equation} - \int_{\bS^n} \omega_n = \frac{2^{m+1}\pi^n}{(2m-1)!}, \quad\mbox{if } n=2m,\; m\geq 1, + \int_{\bS^n} \omega_n = \frac{2^{m+1}\pi^m}{(2m-1)!}, \quad\mbox{if } n=2m,\; m\geq 1, \end{equation} and \begin{equation} @@ -709,3 +709,31 @@ \section{Stokes' Theorem} Prove this corollary. \\ \textit{\small Hint: follows from two the previous corollaries.} \end{exercise} + + +\begin{exercise} +In this problem we are going to prove the smooth version of Brouwer's fixed point theorem. + +\begin{theorem}[Brouwer's fixed point theorem (smooth version)]\label{thm:bfp1} + Let $D_n:=\{x\in\mathbb{R}^n\;\mid\;\|x\|\leq 1\}$ denote the closed unit disk in $\mathbb{R}^n$. % and $\mathbb{S}^{n-1} = \partial D_n$ its boundary, the $(n-1)$-sphere. + Any smooth map $g: D_n \to D_n$ has a fixed point, that is, $\exists\,p\in D_n$ such that $g(p) = p$. +\end{theorem} + +We will proceed by first showing another result. + +\begin{theorem}\label{thm:bfp2} + Let $N$ be a compact $n$-dimensional submanifold of $\mathbb{R}^n$ with non-empty boundary $\partial N$. + Then, there is no differentiable map $f: N \to \partial N$ for which every boundary point is a fixed point, that is, for which $f(p) = p$ for all $p\in\partial N$. +\end{theorem} + +Let $\Omega = dx^1 \wedge \cdots\wedge dx^n$ denote the standard volume form on $N$, that is, the restriction of the standard volume form on $\mathbb{R}^n$ to $N$, +and $X$ be an outward-pointing vector field on $\partial N$. +%Finally, let $i:\partial N\hookrightarrow N$ be the inclusion map. + +\begin{enumerate} + \item Show that $\omega = \iota_X \Omega$ is a closed non-vanishing form on $\partial N$. + \item Show that $f^*\omega$ is closed. + \item Prove Theorem~\ref{thm:bfp2}.\\\textit{Hint: assume there is $f$ such that $f(p)=p$ for all $p\in \partial N$ and use integration to get a contradiction.} + \item Prove Theorem~\ref{thm:bfp1}.\\\textit{Hint: by contradiction, use the half line from $p$ to $g(p)$ to construct a function for which every point in the boundary is fixed. } +\end{enumerate} +\end{exercise} \ No newline at end of file diff --git a/TODO.md b/TODO.md index 0c6a164..275abd4 100644 --- a/TODO.md +++ b/TODO.md @@ -1,11 +1,7 @@ Changes which would complicate something for this year but should be performed for next year: -- move the block of text between Example 1.2.20 and Remark 1.2.23 to some place after Definition 1.3.7 since they involve diffeomorphisms. - - Add theorem showing the link between smooth functions, smooth maps between manifolds and charts. -- Move 2.8.21 at the end of 2.8 - - Explain that vector bundles are useful to extend meaning of vector valued functions, where vector spaces are moving with the point, and mention it is important to understand geometry of injections and topology of spaces. - Add pictures (video?) on how to draw a vector bundle on a cylinder and on a Moebius strip. Link it with the topology justification above. @@ -16,10 +12,6 @@ Changes which would complicate something for this year but should be performed f - Orientation, drop all intro and rewrite it using definition in terms of volume forms. -- Move 5.3.4 in before 5.3.2 - -- Add proof of fixed point thm using Stokes or a detailed exercise about it - - Mention Hodge star operator and maybe Laplace-Beltrami in appendix after volume forms - Add exercise on Liouville theorem for hamiltonians in symplectic geometry diff --git a/aom.tex b/aom.tex index 6603314..c74bc0e 100644 --- a/aom.tex +++ b/aom.tex @@ -208,7 +208,7 @@ \setlength{\parskip}{\baselineskip} Copyright \copyright\ \the\year\ \thanklessauthor - \par Version 0.9.8 -- \today + \par Version 0.10 -- \today \vfill \small{\doclicenseThis}