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\title{Exercises for a geometric introduction to Manifolds}
\author{Roland van der Veen}
\date{Summer 2015}
\begin{document}
\maketitle
\section*{Section 1.2 Preliminaries}
\noindent {\bf Exercise 1} (Polyhedral Gauss-Bonnet)\\
Given a finite set of closed Euclidean triangles $T_1,\dots,T_k\subset\R^3$ the union $S=\cup_i T_i$ is called a polyhedral surface if
the following three conditions hold. No two triangles intersect in more than one of their sides. Every side of $T_i$ belongs to exactly one other triangle $T_j\neq T_i$.
And finally, if $v$ is the common vertex for the triangles $T_{i_1}, T_{i_2},\dots T_{i_n}$ and $s_{i_k}$ is the side opposite to $v$ in triangle $T_{i_k}$ then
the union of the $s_{i_k}$ is connected.
For each vertex $v$ of $S$ we define the curvature $K(v)$ to be $2\pi$ minus the sum of angles directly adjacent to the vertex $v$.
\begin{enumerate}
\item[a.] Show that if $S$ is a regular tetrahedron then $K(v) = \pi$ for each vertex $v$.
\item[b.] Prove that $\sum_{v\in V} K(v) =2\pi \chi(S)$ where $\chi$ is the Euler characteristic $V-E+F=\chi(S)$. The number of edges is $E$ and the number of faces (triangles) is $F$.
This is a polyhedral version of the Gauss-Bonnet theorem where instead of integrating the curvature we concentrate it at the vertices and sum.
\end{enumerate}
\noindent {\bf Exercise 2} (Open sets in vector spaces)\\
Fix an $n$-dimensional vector space $V$. Recall a subset $A\subset V$ is open if it is the image of an open subset of $\R^n$ under some isomorphism $\psi:\R^n\to V$.
\begin{enumerate}
\item[a.] Is $\R-\mathbb{Q}$ an open subset of $\R$?
\item[b.] Is $[0,1]$ an open subset of $[0,3)\subset \R$? What about $[0,1)\cup (2,3)$?
\item[c.] Show that a linear map $L:\R^n\to \R^n$ is continuous.
\item[d.] Prove that if $\phi:\R^n\to V$ is an isomorphism and $A= \psi(B)$ for some open subset $B\subset \R^n$ then $A = \phi(B')$ for some other open subset $B\subset \R^n$.
\item[e.] Conclude that the definition of open subset in $V$ does not depend on the chosen isomorphism $\psi$.
\end{enumerate}
\noindent {\bf Exercise 3} (Linear algebra)\\
\begin{enumerate}
\item[a.] Prove that for $F\in L(U,V)$ and $G\in L(V,W)$ the composition $G\circ F$ is in $L(U,W)$.
\item[b.] Show that $(G\circ F)^*=F^*\circ G^*$.
\item[c.] Compute the dimension of the space $L(V,L(V,W))$.
\item[d.] Prove that in the $n$-dimensional vector space $V$ any linear independent subset can be extended to a basis for $V$.
\end{enumerate}
\noindent {\bf Exercise 4} (Derivative)\\
\begin{enumerate}
\item[a.] Show that for a linear function $F\in L(V,W)$ and any $a\in V$ we have $DF(a)(v) = F(v)$.
\item[b.] Prove that in the definition of derivative the map $\lambda$ is indeed unique.
\item[c.] For $F:\R^2\to\R^3$ given by $F(x,y) = (x^2+2xy,\cos(y),x)$ compute the matrix of $DF(0,\pi)$ with respect to the basis
$b_1=e_1+e_2, b_2 = e_1-e_2$ of $\R^2$ and the basis $e_1,e_2,e_3$ of $\R^3$.
\end{enumerate}
\noindent {\bf Exercise 5} (Smooth inverse, and piecewise functions)\\
\begin{enumerate}
\item[a.] Show that $f(x)=x^3$ is smooth and a bijection $f:\R\to\R$ but its inverse is not smooth.\\
\item[b.] Construct a $C^2$ function $f:\R\to [0,1]$ such that $f|_{[-1,1]} = 1$ and $f(x)=0$ for $|x|>2$.
\end{enumerate}
\noindent {\bf Exercise 6} (Product rule for linear transformations)\\
Imagine differentiable functions $M:\R \to L(V,W)$ and $f:\R\to V$.
\begin{enumerate}
\item[a.] Write down simple examples of $M$ and $f$ in case $V=W=\R^2$.\\
\item[b.] The function that sends $t$ to $M(t)(f(t))$ will be denoted by $Mf(t)$ as if $M$ is a matrix and $f$ a vector.
Show the following product rule holds
\[
D_1 Mf(t) = (D_1M(t)) f(t)+ M(t)(D_1f(t))
\]
\item[c.] Can you give a proof of this rule without chooing a basis for $V$ and $W$?
\end{enumerate}
\section*{Section 2.1 Manifolds}
\noindent {\bf Exercise 1} (Compactness)\\
\begin{enumerate}
\item[a.] Find two coordinate patches that cover the circle.\\
\item[b.] Prove that one coordinate patch will never cover the circle. Hint use compactness.
\item[c.] Use a similar argument to show that the closed interval $[0,1]$ is not a manifold.
\item[d.] Is the unit square $(0,1)\times (0,1)$ a manifold?
\end{enumerate}
\noindent {\bf Exercise 2} (A special case of the rank theorem)\\
In this exercise we explore the proof of the rank theorem in a special case, take $f:\R^2\to\R$ and assume
$Df(x,y)$ has rank $1$ for every $(x,y)\in\R^2$. We would like to prove that $X = f^{-1}\{0\}$ is a $1$-manifold in $\R^2$.
We assume $X$ is non-empty.
\begin{enumerate}
\item[a.] Explain why we may assume without loss of generality that $D_1 f(x,y)\neq 0$.\\
\item[b.] Define $F:\R^2\to \R^2$ by $F(x,y) = f(x,y)e_1+y$. With the above assumption, show that
$\det DF(x,y) \neq 0$.
\item[c.] Apply the Inverse function theorem to $F$ to show the following. For any $p = \in \R^2$ there is
an open set $A\ni F(p)$ and a function $G:A\to B \ni p$ between open subsets of $\R^2$ such that $G\circ F = id_B$ and $F\circ G = id_A$.
\item[d.] Now take $p\in f^{-1}\{0\}$ so $f(p) = 0$. Show that restricted to $A\ni p$ the map $F$ is a diffeomorphism between
$A$ and $B$ as above and $F(X\cap A) = B\cap (\{0\}\times \R)$.
\item[e.] Conclude that $X\cap A$ is a coordinate patch for $X$ containing $p$. What is the formula for the corresponding parameterization?
What is the chart?
\item[f.] Taking the familiar $f(x,y) = x^2+y^2-1$, show that this construction yields the usual coordinate patches on the circle.
\end{enumerate}
\noindent {\bf Exercise 3} (Building $C^k$ surfaces in $\R^3$)\\
Here we examine in what sense the four pieces produce a $C^k$ surface in $\R^3$.
\begin{enumerate}
\item[a.] Show that the function $S(x,y)$ is actually $C^k$ on the unit square.
\item[b.] The four pieces may be varied in that we can permute the three coordinates
and may replace any coordinate $t$ by $1-t$. We may also translate them by adding vectors in $\Z^3$.
Explain why the resulting pieces fit together in a $C^k$ way if at all.
\item[c.] Build an elephant using the building blocks.
\end{enumerate}
\noindent {\bf Exercise 4} (Examples of manifolds)\\
\begin{enumerate}
\item[a.] Given manifolds $X\subset V$ and $Y\subset W$, prove that $X\times Y\subset V\times W$ is a manifold too.
\item[b.] For some large number $R$ define the M\"{o}bius strip by $M_R = \mathrm{im}f$ where $f:(-1,1)\times[0,2\pi)$ is given by
$f(r,t) = \cos t(R+r\cos \frac{t}{2})e_1+\sin t(R+r\cos \frac{t}{2})e_2+r\sin\frac{t}{2}e_3$. Show that $M_R$ is a manifold for $R$ sufficiently large.
\item[c.] Is $X = \{(x,y,z,w)\in \R^4| x^4+y^2-w^2=1, e^z+e^w= 1 \}$ a manifold? If so of what dimension?
\item[d.] Viewing $\mathbb{C}^2$ as an 4-dimensional vector space over $\R$, prove that the graph of an entire holomorphic
function is a $2$-manifold.
\end{enumerate}
\noindent {\bf Exercise 5}\\
Try to prove the last two lemmas.
\section*{Section 2.2 Tangent Space}
\noindent {\bf Exercise 1} (Definition of the Tangent space)\\
Imagine a manifold $X\subset V$ and a coordinate patch $\phi:B\to C\subset X$. Here $B\subset W$ is some
open subset of vector space $W$.
Recall we defined the tangent space $T_xX$ as
the image $\mathrm{im}D\phi(b)$ where $\phi(b)=x\in C$.
\begin{enumerate}
\item[a.] Suppose we have another coordinate patch $\phi':B'\to C'\ni x$ such that $\phi'(b')=x$.
Show that $F=\phi^{-1}\circ \phi'$ is a diffeomorphism from $\phi'^{-1}(C\cap C')$ to $\phi^{-1}(C\cap C')$
and $DF(b'):W\to W$ is a linear isomorphism.
\item[b.] Show that $D\phi'(b') = D\phi(b)\circ DF(b')$.
\item[c.] Explain why the equation in part b. implies $\mathrm{im}D\phi'(b') = \mathrm{im}D\phi(b)$.
Conclude the definition of $T_xX$ does not depend on the chosen coordinate patch.
\end{enumerate}
\noindent {\bf Exercise 2} (Tangent spaces)\\
Here we explore some elementary examples of tangent spaces.
\begin{enumerate}
\item[a.] Describe the tangent space $T_xX$ where $X\subset V$ is an open subset of vector space $V$.
\item[b.] Describe the tangent space $T_xX$ to the graph $X\subset \R^2$ of a differentiable function $f:\R\to \R$ at the point $x = (t,f(t))$.
\item[c.] Suppose $f:V\to \R$ is a differentiable map of rank $1$ sending $0$ to $0$. Prove that $T_xX = \mathrm{ker} Df(x)$ where
$X = f^{-1}(0)$ is (by the rank theorem) a hypersurface.
\end{enumerate}
\noindent {\bf Exercise 3} (M\"{o}bius strip)\\
The M\"{o}bius strip $M$ was discovered by the the projective geometer Ferdinand M\"{o}bius as the thing that remains when one deletes a disk from the projective plane.
We will realize $M$ as a $2$-manifold in $\R^3$ as the image of the function $F:\R\times (-\frac{1}{2},\frac{1}{2})\to \R^3$ defined by
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{Pictures/Moeb.png}
\end{center}
\end{figure}
\[F(\theta,t) = h(\theta)+t \left(\cos\left(\frac{\theta}{2}\right)h(\theta)+\sin\left(\frac{\theta}{2}\right)e_3\right)\]
where $h(\theta) = (\cos \theta) e_1+(\sin\theta) e_2$.
\begin{enumerate}
\item[a.] Before getting started we would like to see that $M$ actually is a manifold. To this end,
show that $\phi=F|_{(-\pi,\pi)\times (-\frac{1}{2},\frac{1}{2})}$ and $\psi=F|_{(0,2\pi)\times (-\frac{1}{2},\frac{1}{2})}$
are parameterizations for $M$ whose images cover $M$.
\item[b.] What is the dimension of $M$?
\item[c.] Give a basis for the tangent space at the point $F(\theta,0)$ for some fixed $\theta\in [0,2\pi)$,
such that the first basis-vector is in the horizontal $xy$ plane spanned by $e_1$ and $e_2$.
\item[d.] Something odd happens in part c that has to do with the non-orientability of $M$. Can you guess what it is?
We will come back to this feature later.
\item[e.] Define the function $g:M\to \R^2$ by $g(x,y,z) = e^{xyz}e_1+xe_2$.
Give a matrix for the linear transformation $Dg(1,0,0)$ with respect to a basis of your choice.
\end{enumerate}
\section*{Section 2.3 Geodesics}
\noindent {\bf Exercise 1} (Geodesics in a vector space)\\
Describe all geodesics in a vector space $V$.\\
\noindent {\bf Exercise 2} (Geodesics on a cylinder)\\
Write down and solve the geodesic equation for the cylinder
\[X = \{(x,y,z)\in \R^3| x^2+y^2=1\}\]
\section*{Section 2.4 Vector fields}
\noindent {\bf Exercise 1} (Existence and Uniqueness on manifolds)\\
Prove the following theorem by following the steps outlined below:\\
\begin{theorem}
For any differentiable tangent vector field $F:X\to V$ on a manifold $X\subset V$ and any $x\in X$ there exists an integral curve $\gamma$ such that $\gamma(0) = x$.
Any two such curves with the same domain must coincide.
\end{theorem}
\begin{enumerate}
\item[a.] Choose a coordinate patch $\phi:B\to C\subset X$ such that $x\in C$ where $B$ is some open subset of a vector space $W$. Why is
$G:B\to W$ given by $G(b) = (D\phi(b))^{-1}F(\phi(b))$ a well defined and differentiable vector field?
\item[b.] Apply the ODE theorem for existence and uniqueness of integral curves to obtain an integral curve $\alpha:[-\epsilon,\epsilon]\to B$ for $G$
with $\alpha(0) = b$.
\item[c.] Use the chain rule to prove that $\gamma = \phi\circ \alpha$ is an integral curve for $F$ and explain why this proves the existence part of our theorem.
\item[d.] Finally prove the uniqueness by again reducing it to the case in $B$.
\end{enumerate}
\noindent {\bf Exercise 2} (Special vector field)\\
Give a formula for a vector field $F:\R^2\to \R^2$ with the following properties. $F$ should be differentiable and $(0,0)$ should be the only point where
$F=0$. Also for any fixed $r>0$ and $t$ running from $0$ to $2\pi$ the unit vectors $\frac{F(r\cos t,r\sin t)}{|F(r\cos t,r\sin t)|}$ should rotate around the unit circle
twice in the counter clockwise direction. You dont have to prove your claim, giving a formula is sufficient. Hint: check out the hints page on the website.\\
\section*{Section 2.5 Curvature}
\noindent {\bf Exercise 1} (Computing curvature)\\
For any $a\in \R$ define $X= \{ (x,y,z)\in \R^3| z= x^2+ay^2\}$
\begin{enumerate}
\item[a.] Show that $X$ is a $2$-manifold in $\R^3$ parametrized by $\phi:\R^2\to X$ defined by $\phi(x,y) = xe_1+ye_2+(x^2+ay^2)e_3$.
\item[b.] Find a basis for the tangent space $T_pX$ at point $p = (x,y,z)$.
\item[c.] Write down an equation for a (unit) normal vector field $\N:X\to\Sp^2$.
\item[d.] Explain why $T_p X = T_{\N(p)}\Sp^2$ for any point $p$. (hint look at the orthogonal complement).
\item[e.] Give the matrix of $D\N(0,0,0)$ with respect to the basis from part b.
\item[f.] Compute the curvature $\kappa(0,0,0)$ of $X$ at the point $(0,0,0)$.
\end{enumerate}
\section*{Section 2.6 Multilinear algebra, differential forms and orientation}
\noindent {\bf Exercise 1} (Computing curvature)\\
Referring to the previous exercise 2.2.3 on the M\"{o}bius strip, prove that it can not be orientable. Hint:
Assume an orientation $\omega$ exists and consider what happens if you evaluate $\omega$ on the smoothly varying
bases you found in part c.
\\
\noindent {\bf Exercise 2} (Dimension of $\Alt^n V$)\\
Prove that if $V$ is $n$ dimensional, $\Alt_n(V)$ is a vector space and its dimension is $1$. Bonus:
what's the dimension of $\Alt^k(V)$?\\
\noindent {\bf Exercise 3} (Independent alternating maps on $\R^4$)\\
Write down two independent elements of $\Alt^2(\R^4)$.\\
\end{document}