Add some remarks, reference and exercises

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

1-manifolds.tex

@@ -129,22 +129,25 @@ \section{Topological manifolds}
   More generally, any $n$-dimensional vector space\footnote{In fact, any open subset of a $n$-dimensional vector space.} is a topological $n$-manifold.
 \end{example}
 
-\begin{exercise}
-	Even though $\R^n$ with the euclidean topology is Hausdorff, being Hausdorff does not follow from being locally euclidean. A famous counterexample is the following\footnote{See also Problem 5.1 in~\cite{book:tu}.}.
+\begin{exercise}[The line with two origins]
+	Even though $\R^n$ with the euclidean topology is Hausdorff, being Hausdorff does not follow from being locally euclidean. A famous counterexample is the following\footnote{See also \cite[Problem 1-1]{book:lee} and \cite[Problem 5.1]{book:tu}.}.
   \begin{marginfigure}
     \includegraphics{1_ex_1_0_11.pdf}
     \label{fig:hausdorff-not-locally-euclidean}
     \caption{A locally euclidean space which is not Hausdorff.}
   \end{marginfigure}
-  Let $A_1, A_2$ be two point not on the real line $\R$ and define $M:= (\R\setminus\{0\})\cup\{A_1,A_2\}$.
-  Define the two charts
-  \begin{equation}
-  \varphi_j:(\R\setminus\{0\})\cup\{A_j\} \to \R, \quad
-  \varphi_j(x) = \begin{cases} x &\mbox{if } x\neq A_j\\ 0 & \mbox{if } x = A_j \end{cases}, \quad
-  j = 1,2.
-  \end{equation}
+  Let $P, Q$ be two point not on the real line $\R$ and define $M:= (\R\setminus\{0\})\cup\{O_1,O_2\}$.
+  Induce a topology on $M$ by taking as basis the collection of all open intervals in $\R$ that do not contain $0$, along with all the sets of the form $(-a, 0)\cup\{O_1\}\cup(0,a)$ and $(-a, 0)\cup\{O_2\}\cup(0,a)$, for $a>0$.
   \begin{enumerate}[(a)]
-    \item Show that $\varphi_1$ and $\varphi_2$ are homeomorphisms with respect to the topology induced by the two charts\footnote{Let $(X, \cT)$ be a topological space and $f: X\to Y$ some map. The induced topology on $Y$ is \begin{equation}\cU_f := \{f^{-1}(U) \;\mid\; U\in\cT\}.\end{equation}}.
+    \item Check that this forms a basis\footnote{That is, the basis elements cover $M$ and for any $B_1, B_2$ on the basis, for all $x \in I = B_1\cap B_2$, there is an element $B_3$ of the basis such that $x\in B_3$ and $B_3\subset I$.} for a topology on $M$.
+    \item Define the two charts
+    \begin{equation}
+    \varphi_j:(\R\setminus\{0\})\cup\{O_j\} \to \R, \quad
+    \varphi_j(x) = \begin{cases} x &\mbox{if } x\neq O_j\\ 0 & \mbox{if } x = O_j \end{cases}, \quad
+    j = 1,2.
+    \end{equation}
+    Show that $\varphi_1$ and $\varphi_2$ are homeomorphisms with respect to the aforementioned topology. 
+    %induced by the two charts\footnote{Let $(X, \cT)$ be a topological space and $f: X\to Y$ some map. The induced topology on $Y$ is \begin{equation}\cU_f := \{f^{-1}(U) \;\mid\; U\in\cT\}.\end{equation}}.
     \item Show that $M$ is locally euclidean and second countable but not Hausdorff.
   \end{enumerate}
 \end{exercise}
@@ -337,6 +340,11 @@ \section{Differentiable manifolds}
   \end{exercise}
 \end{example}
 
+\begin{exercise}
+  Let $\{(U_\alpha, \varphi_\alpha)\}$ be the maximal atlas on a manifold $M$.
+  For any open set $U\subseteq M$ and any point $p\in U$, prove the existence of a coordinate open set $U_\alpha$ such that $p\in U_\alpha\subset U$.
+\end{exercise}
+
 \begin{exercise}
   Let $f: \R^n \to \R^m$ be a smooth map.
   Show that its graph
@@ -348,7 +356,7 @@ \section{Differentiable manifolds}
 
 \begin{example}
   The definition of smooth manifold does not require $M$ to be embedded into some ambient space as in the examples above.
-  In fact, we can define the differentiable manifold $\bS^1$ by equipping the topological quotient space\footnote{
+  In fact, we can define the differentiable manifold $\bS^1$ by equipping the topological quotient space\sidenote[][-11em]{
     There is a standard way to induce a topology on a quotient space.
     Let $M$ be a topological space and $\pi:M\to N$ surjective.
     The \emph{quotient topology} on $N$ is given by defining $U\subset N$ to be open if and only if its preimage $\pi^{-1}(U)\subset M$ is open.
@@ -381,6 +389,23 @@ \section{Differentiable manifolds}
   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}
+  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 $\cA_s$, $0<r<s$, are different.
+    This shows that there are uncountably many families of different differential structures on $\R$.
+    \item Let $M_r$ be the manifold $\R$ equipped with the atlas $\cA_r$.
+    Show that $M_r$ and $M_s$ are diffeomorphic for $r,s >0$.
+  \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.}.
@@ -425,6 +450,16 @@ \section{Differentiable manifolds}
 If $M$ is a topological space and $\sim$ an equivalence relation we have seen that it is sometimes possible to define smooth manifolds.
 Since in general the quotient does not behave nicely it is convenient to get a few tricks to check if the manifold structure can be preserved.
 
+In this case it is convenient to have some tools to check continuity of functions.
+
+\marginnote{For a proof refer to \cite[Proposition 7.1]{book:tu} or \cite[Theorem 3.70]{book:lee:topology}.}
+\begin{proposition}
+  Assume $F:X\to Y$ is a map between topological spaces and $\sim$ is an equivalence relation on $X$.
+  Let $F$ be constant on each equivalence class $[p]\in X/\!\sim$, and denote $\widetilde F:X/\!\sim\to Y$, $\widetilde F([p]) := F(p)$ for $p\in X$, the map induced by $F$ on the quotient.
+
+  Then, $\widetilde F$ is continuous if and only if $F$ is continuous.
+\end{proposition}
+
 Continuity of the projection implies that if $M/\!\sim$ is Hausdorff, then $\pi^{-1}(\pi(s)) = [s]$ is closed in $M$.
 If, additionally, $\pi$ is open\footnote{That is, it maps open sets to open sets.} then there is a stronger statement:
 \marginnote[4em]{These statements are not hard to prove, but their proofs will be omitted here.
@@ -471,7 +506,7 @@ \section{Differentiable manifolds}
   This leads to the bijection $\RP^n \simeq \bS^n/\!\sim$.
   Note that by gluing antipodal points, we are identifying the north and south hemispheres, thus essentially flattening the sphere to a disk.
 
-  \begin{exercise}
+  \begin{exercise}\label{exe:RPSN}
     Show that the map $n: \R^{n+1}_0\to \bS^n$, $n(x) = \frac{x}{\|x\|}$ induces a homeomorphism $\hat n:\RP^n \to \bS^n/\!\sim$.\\
     \textit{\small Hint: find an inverse map and show that both $\hat n$ and its inverse are continuous.}
   \end{exercise}
@@ -513,6 +548,11 @@ \section{Differentiable manifolds}
   The atlas defined by the collection $\{(U_i, \varphi_i)\}$ is called \emph{standard atlas} and makes $\RP^n$ a smooth manifold. 
 \end{example}
 
+\begin{exercise}
+  Show that the real projective space $\RP^n$ is compact.\\
+  \textit{\small Hint: use Exercise~\ref{exe:RPSN}.}
+\end{exercise}
+
 \begin{exercise}\label{ex:stereo}
   Let $N$ denote the north pole $(0,\ldots,0,1)\in\bS^n\subset\R^{n+1}$ and let $S$ denote the south pole $(0,\ldots,0,-1)$.
   Define the \emph{stereographic projections} $\sigma:\bS^n\setminus\{N\}\to \R^n$ by
@@ -570,7 +610,23 @@ \section{Smooth maps and differentiability}
   Then, the space $C^\infty(M)$ endowed with the operations above is an \emph{algebra}\footnote{I.e. a vector space where you can also multiply two elements.} over $\R$.
 \end{exercise}
 
-Which immediately gives away the general definition.
+The following theorem can be very convenient when you work with smooth functions.
+
+\begin{proposition}
+  Let $M$ be a smooth $n$-manifold and $f:M\to\R$ a real-valued function on $M$. Then, the following are equivalent:
+  \begin{enumerate}[(i)]
+    \item $f\in C^\infty(M)$;
+    \item $M$ has an atlas $\cA$ such that for every chart $(U, \varphi)\in\cA$, $f\circ \varphi^{-1} : \R^n\supset\varphi(U)\to \R$ is $C^\infty$;
+    \item for every chart $(V,\psi)$ on $M$, the function $f\circ\psi^{-1} : \R^n\supset\psi(U)\to \R$.
+  \end{enumerate}
+\end{proposition}
+
+\begin{exercise}
+  Prove the proposition.\\
+  \textit{\small Hint: go cyclic, for example show $(ii)\Rightarrow(i)$, $(i)\Rightarrow(iii)$, $(iii)\Rightarrow(ii)$.}
+\end{exercise}
+
+At this point, the generalization of smooth functions to smooth maps between manifolds should not come as a surprise.
 
 \begin{definition}
   Let $F:M_1\to M_2$ be a continuous map \footnote{Remember: continuity is not a problem since $M_1$ and $M_2$ are topological spaces.} between two smooth manifolds of dimension $n_1$ and $n_2$ respectively.
@@ -625,16 +681,19 @@ \section{Smooth maps and differentiability}
     Let $M, N, P$ be three smooth manifolds, and suppose that $F:M\to N$ and $G:N\to P$ are smooth.
     Then $G\circ F\in C^\infty(M, P)$.
   \end{proposition}
-  \marginnote{In Proposition~\ref{prop:uniqdiffeoinclusion} it is not enough to ask that $\iota$ is smooth! As counterexample consider the two manifolds $(\R, \cA_1)$ with $\cA_1 := \{(\R, \id_\R)\}$ and $(\R, \cA_2)$ with $\cA_2 := \{(\R, x\mapsto x^3)\}$. The inclusion of open sets in $\R$ is smooth in both cases but is a diffeomorphism only in one.}
-  \begin{proposition}\label{prop:uniqdiffeoinclusion}
-    Let $M$ be a manifold and $U\subset M$ an open set.
-    Then $U$ has a unique differentiable structure such that the inclusion $\iota:U\hookrightarrow M$ is a diffeomorphism.
-  \end{proposition}
+
   \begin{proposition}[Smoothness is a local property]\label{prop:smoothlocal}
     Let $F:M\to N$ be a continuous function and let $\{U_i\}_{i\in I}$ be an open cover for $M$. Then $F|_{U_i}:U_i \to N$ is smooth for every $i\in I$ iff $F:M\to N$ is smooth.
   \end{proposition}
 \end{exercise}
 
+\begin{exercise}
+  Prove that $\R^2\setminus\{(0,0)\}$ is a two-dimensional manifold and construct a diffeomorphism from this manifold to the circular cylinder
+  \begin{equation}
+    C := \{ (x,y,z)\in\R^3 \mid x^2+y^2 = 1\}\subset\R^3.
+  \end{equation}
+\end{exercise}
+
 The following corollary is just a restatement of Proposition~\ref{prop:smoothlocal}, but provides a useful perspective on the construction of smooth maps.
 
 \begin{proposition}[Gluing lemma for smooth maps]

2-tangentbdl.tex

@@ -1088,6 +1088,14 @@ \section{Submanifolds}
   Finally, with the inclusion $i:M\hookrightarrow N$ one has that $\psi \circ i\circ \sigma^{-1} (p^1,\ldots,p^n) = (p^1,\ldots,p^n,0,\ldots,0)$ which is smooth.
 \end{proof}
 
+A non-trivial consequence of the previous lemma is the following proposition\footnote{Refer to \cite[Proposition 5.8 and Proposition 5.31]{book:lee}.}.
+
+\marginnote{In Proposition~\ref{prop:uniqdiffeoinclusion} it is not enough to ask that $\iota$ is smooth! As counterexample consider the two manifolds $(\R, \cA_1)$ with $\cA_1 := \{(\R, \id_\R)\}$ and $(\R, \cA_2)$ with $\cA_2 := \{(\R, x\mapsto x^3)\}$. The inclusion of open sets in $\R$ is smooth in both cases but is a diffeomorphism only in one.}
+\begin{proposition}\label{prop:uniqdiffeoinclusion}
+  Let $M$ be a manifold and $U\subset M$ an open set.
+  Then $U$ has a unique differentiable structure such that the inclusion $\iota:U\hookrightarrow M$ is a diffeomorphism.
+\end{proposition}
+
 Up to this point, the first manifold had the same dimension or was smaller than the second one.
 What if it is larger?
 
@@ -1150,6 +1158,15 @@ \section{Submanifolds}
   For example with the global atlas $\{(M,\; (x^1,x^2)\mapsto x^1)\}$, $M$ is a manifold diffeomorphic to $\R$.
 \end{example}
 
+\begin{exercise}
+  A real-valued function $f:M\to\R$ on a manifold has a local maximum at $p\in M$ if there is a neighbourhood $U\subset M$ of $p$ such that $f(p) \geq f(q)$ for all $q\in U$.
+  \begin{enumerate}
+    \item Show that if a differentiable function $f:(a,b)\to\R$, has a local maximum at $x\in (a,b)$, then $f'(x) = 0$.
+    \item Prove that a local maximum of a function $f\in C^\infty(M)$ is a critical point of $f$.\\
+    \textit{\small Hint: choose $X_p\in T_pM$ and let $\gamma(t)$ be a curve in $M$ starting at $p$ with initial velocity $X_p$. The $f\circ \gamma$ is a real-valued function with local maximum at $0$...}
+  \end{enumerate}
+\end{exercise}
+
 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.

3-vectorfields.tex

@@ -163,6 +163,11 @@ \section{Vector fields}
   If $X\in\fX(M)$ and $f\in C^\infty(M)$, then $Xf\in C^\infty(M)$.
 \end{exercise}
 
+\begin{exercise}
+  Let $X,Y\in\fX(M)$.
+  Show that $X=Y$ if and only if $Xf = Yf$ for every $f\in C^\infty(M)$.
+\end{exercise}
+
 This whole discussion allows us to extend the notion of derivation at a point to a derivation on the whole space.
 \marginnote[2em]{Don't confuse this with the derivations at a point, which produce real numbers. In this case we map functions to functions. }
 \begin{definition}

4-cotangentbdl.tex

@@ -329,6 +329,16 @@ \section{One-forms and the cotangent bundle}
 \end{proof}
 In this case you often say that the vector fields are $F$-related or that they behave naturally: you can either pull back the function $f$ to $M$ or push forward the vector field $X$ to $N$.
 
+\begin{exercise}
+  Let $\{\rho\alpha\}$ denote a partition of unity on a manifold $M$ subordinate to an open cover $\{U_\alpha\}$.
+  Let $F:N\to M$ denote a smooth map between smooth manifolds.
+  With the definition of pullback of functions given above, prove that
+  \begin{enumerate}[(a)]
+    \item the collection of supports $\{\supp F^*\rho_\alpha\}$ is locally finite;
+    \item the collection of functions $\{F^*\rho_\alpha\}$ is a partition of unity on $N$ subordinate to the open cover $\{F^{-1}(U_\alpha)\}$ of $N$.
+  \end{enumerate}
+\end{exercise}
+
 When we discussed vector fields, we observed that pushforwards of vector fields under smooth maps are defined only in the special case of diffeomorphisms.
 The surprising thing about covectors is that covector fields always pull back to covector fields.
 

5-tensors.tex

@@ -1,3 +1,4 @@
+\marginnote{For a brief and \emph{concrete} explanation of tensors, I warmly recommend the following \href{https://youtu.be/f5liqUk0ZTw}{a youtube video by Dan Fleisch}.}
 Many of the spaces that we have encountered so far are particular examples of a much larger class of objects.
 In this chapter we are going to introduce all the necessary algebraic concepts.
 

6-differentiaforms.tex

@@ -34,6 +34,7 @@ \section{Differential forms}
 
 \section{The wedge product}
 
+\marginnote{You can find an interesting explanation of the wedge product, based on Penrose's book ``The road to reality'', \href{https://twitter.com/LucaAmb/status/1289244374996406273?s=20}{on a thread by @LucaAmb on Twitter}.}
 If you remember, we said that the determinant was an example of a $T_n^0(R^n)$ tensor: an antisymmetric tensor nonetheless.
 At the same time, the determinant of a $n\times n$ matrix, is the signed volume of the parallelotope spanned by the $n$ vectors composing the matrix.
 We also saw that tensors can be multiplied with the tensor product, which gives rise to a graded algebra on the free sum of tensor spaces.