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\begin{document}
\title{{\bf  Homework set 1 - due 09/06/24}\\ {\large Math 5047} 
}
\author{}
\date{}  
\maketitle
\thispagestyle{empty}
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 \begin{itemize}
\item For most of my problems, answers will consist of pointing to a theorem or theorems and  performing some calculation.  I mainly want to see that you can do the calculations and are aware of the pertinent theorems.  Try to be clear and succinct in your writing. (The grader will be grateful!)

\item Begin to work on  these assignments early and feel free to ask questions about them in class. I can't be sure of what you'd have seen in 
the first two semesters of Geometry/Topology, or even whether I'm using familiar notation. Always feel free to bring up for class discussion whatever seems confusing or possibly related to material I may be assuming you know that was not covered in Geometry/Topology I and II.

\item Assignments will typically be due Friday at 11:59PM. They will handled via Gradescope. (I'm not familiar with it myself! I hope to figure it out before this assignment's due date.)
Whenever possible, I plan to take a few minutes from class every Wednesday to discuss the assignment due Friday. Make an effort to start working on it early to make the Wednesday discussion most helpful to you.


\item Homework sets will contain a variable number of problems, but only 4 will actually be graded. (I don't want to overwhelm the grader with work.) Solutions to all of them will be available after the due date.

\item The final exam will be based on material taken from  homework assignment questions.  
 \end{itemize}

\begin{center}
{\bf Problems}
\end{center}

\begin{enumerate}

\item Do a careful browsing of the appendices A, B, C of Lee's text.   


\item
Using the standard notation $(x,y,z)$ for the Cartesian coordinates on $\mathbb{R}^3$, 
let $V$ be the complement in $\mathbb{R}^3$ of the half-plane defined by $y=0$  and  $x\geq 0$: 
Let $U$ be the open box $ (0,\infty)\times (0,2\pi)\times (0,\pi)$, with
coordinates   $(\rho, \theta, \varphi)$. The   spherical coordinates on $\mathbb{R}^3$
are given by the following $f$:
\begin{align*}
x&=f_1(\rho,\theta,\varphi)=\rho\sin\varphi\cos\theta\\
y&=f_2(\rho,\theta,\varphi)=\rho\sin\varphi\sin\theta\\
z&=f_3(\rho,\theta,\varphi)=\rho\cos\varphi.
\end{align*}


Find an expression for $f_*\frac{\partial}{\partial \rho}$, $f_*\frac{\partial}{\partial \theta}$ and $f_*\frac{\partial}{\partial \phi}$ in   the coordinates
$(x,y,z)$.


 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item 
The sets   $$SL(n,\mathbb{R})=\{A\in M(n,\mathbb{R}): \text{det} A= 1\}, \ \ 
 O(n)=\{A\in M(n,\mathbb{R}):A^\intercal A=I\}$$ are subgroups of the general linear group $GL(n,\mathbb{R})$ of
invertible real matrices of size $n$-by-$n$. They are called, respectively, the {\em special linear} and the {\em orthogonal} groups.
The intersection $SO(n)=SL(n,\mathbb{R})\cap O(n)$ is called the {\em special orthogonal} group. 
\begin{enumerate}
\item 
Show that $SL(n,\mathbb{R})$ is 
a Lie group. For this, first show that the differential of the determinant function $\text{det}:M(n,\mathbb{R})\rightarrow \mathbb{R}$ 
at any non-singular (i.e., invertible) matrix $A$ is surjective.
 \item  Show that $O(n)=\{A\in M(n,\mathbb{R}):A^\intercal A=I\}$ is a Lie group.
For this, first show that the map $f(A)=A^\intercal A$ from $M(n,\mathbb{R})$ into the space of symmetric, positive,
$n$-by-$n$ matrices is a submersion at each $A\in O(n)$. 
\item  Explain  that $SO(n)=O(n)\cap SL(n,\mathbb{R})$
is a Lie group.   
\end{enumerate}

\item 
The $n$-dimensional  sphere is the set $$S^n=\{q\in \mathbb{R}^{n+1}:\|q\|=1\}.$$
\begin{enumerate}
\item Show that $S^n$ is a smooth submanifold of $\mathbb{R}^{n+1}$.
Do this by showing that for all non-zero $q$ the differential at $q$ of the map $f:\mathbb{R}^{n+1}\rightarrow \mathbb{R}$
defined by $f(q)=\|q\|^2$ is surjective. 
\item Explain why  the quotient  $SO(n+1)/SO(n)$ admits a  smooth manifold structure with respect to which the projection map
$SO(n+1)\rightarrow SO(n+1)/SO(n)$ is smooth. 
\item The special orthogonal group $SO(n+1)$ acts smoothly on the sphere $S^n$  according to the action map $(A,x)\mapsto Ax$. 
 Let $\mathcal{N}=(0,\dots, 0,1)^\intercal$ denote the north pole. We may regard $SO(n)$ as the subgroup 
 $$ SO(n)=\{A\in SO(n+1): A\mathcal{N}=\mathcal{N}\}.$$ Now define a map
 $\varphi:SO(n+1)/SO(n)\rightarrow S^n$ by $\varphi: g SO(n)\mapsto g\mathcal{N}$. Show that this
 map is a diffeomorphism.
\end{enumerate}


\item Consider vector fields on $\mathbb{R}^n$ of the form
$X_A(p)= -Ap $ where $A=(a_{ij})$ is an $n\times n$ real matrix. Here we regard $p$ as a column vector. These are called {\em linear vector fields}.
 In terms of the basis $\{\partial_1,\dots ,\partial_n\}$, where $\partial_i=\frac{\partial}{\partial x_i}$,  $X$ can be written as follows:
$$X_A(p)=-\sum_{ij} a_{ij} x_j(p)\partial_i.$$


Given  two $n\times n$ matrices $A$ and $B$, show that 
$$ [X_A,X_B]= X_{[A,B]}$$
where $[A,B]=AB-BA$ is the matrix  commutator. 

\item 
Recall that $O(n)$ is the orthogonal group in dimension $n$, which consists of
$n$-by-$n$ real matrices $R$ satisfying $R^\intercal=R^{-1}$.
 We already know that $O(n)$ is a Lie group relative to
matrix multiplication and inverse.
Let $$\mathfrak{o}(n)=\{A\in M(n,\mathbb{R}): A^\intercal=-A\}.$$ For each $A\in \mathfrak{o}(n)$
define the vector field on $M(n,\mathbb{R})$ expressed in the coordinates $x=(x_{ij})$ by
$$X_A(x)= \sum_{ij} (xA)_{ij} \frac{\partial}{\partial x_{ij}}.$$
\begin{enumerate}
\item Check that $\mathfrak{o}(n)$, with the matrix commutator $[A,B]=AB-BA$, is a Lie algebra.
\item Show that $[X_A, X_B]= X_{[A,B]} $ for all  $A, B\in \mathfrak{o}(n)$.
\item Show that the flow associated to $X_A$ is given by
$\Phi_t(g)=ge^{tA} $
for all $g\in O(n)$. 
\item Show that   $X_A(g)\in T_g O(n)$ for all $g\in O(n)$.     
\item Show that $X_A$ is a left-invariant vector field on $O(n)$. For this, show that $$(dL_g)_h X_A(h)= X_A(gh).$$
\item Conclude that the Lie algebra of left-invariant vector fields on $O(n)$ is isomorphic to $\mathfrak{o}(n)$ with
the commutator bracket of matrices.
\end{enumerate}






\item Let $M=\mathbb{R}^n$.
Let $Z$ be a smooth vector field on $M$. At every $p\in M$ let $$\Pi_p:T_pM \rightarrow T_pM$$
denote  the orthogonal projection to $Z_p^\perp=\{v\in T_pM: Z_p\cdot v=0\}$.
 It is not difficult to check that $\Pi$
is a  smooth tensor field on $M$ of type $(1,1)$.  Now define
$\tau:\mathfrak{X}(M)\times \mathfrak{X}(M)\rightarrow C^\infty(M)$,
$$ \tau(X,Y)=Z\cdot [\Pi X, \Pi Y].$$
Here $\cdot$ is the ordinary dot product in $\mathbb{R}^n$. 
In other words,  at each point $p$, $\tau(X,Y)$ computes the orthogonal projection to the subspace of $T_pM$ spanned by $Z_p$
of the Lie bracket of the vector fields $\Pi X$ and $\Pi Y$ (both everywhere orthogonal to $Z$).
Show that $\tau$ is a smooth  tensor field of type $(0,2)$. We also say that $\tau$ is {\em tensorial} in the arguments $X, Y$. 

\item Show that if $f:M\rightarrow N$ is a diffeomorphism,
$\theta$  a smooth tensor  field of type $(0,k)$ on $M$, and   $X_1, \dots, X_k\in \mathfrak{X}(M)$, then 
$f_*(\theta(X_1, \dots, X_k))=(f_*\theta)(f_*X_1, \dots, f_*X_k). $ 

(Recall that $f_*h=h\circ f^{-1}$ for $h\in C^\infty(M)$.) 





\item Let $X_1, \dots, X_n$ be everywhere linearly independent and commuting smooth vector fields defined on an 
 open set $U\subset M$ of a smooth, $n$-dimensional  manifold $M$. (There is no loss of generality in 
 assuming that $M=\mathbb{R}^n$.) Let $\Phi_{t_i}^{X_i}$ be the local flow of $X_i$.
We assume that $V\subset U$ is an open subset and $I=(-a,a)$ is a small enough interval such that
$\Phi_{t_i}^{X_i}(p)$ is defined for all $p\in V$ and 
all  $(t_1, \dots, t_n)\in I^n$.
For a fixed $p_0\in V$  define $$ \varphi(t_1, \dots, t_n):=\left( \Phi_{t_1}^{X_1}\circ\cdots \circ \Phi_{t_n}^{X_n}\right)(p_0).$$
Show that $\varphi$ is a diffeomorphism from $I^n$ onto its image in $V$ 
such that $\varphi^*X_i=\frac{\partial}{\partial t_i}.$ Therefore,   the $X_i$ are
the coordinate vector fields of the coordinate neighborhood of $p_0$ defined by $\varphi$.

 \end{enumerate}
 \end{document}
 
 
 
  
 \vspace{.1in}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1.5in]{skrew.eps}
\caption{\small  A screw motion in $\mathbb{R}^3$ consists of a rotation by $\omega t$ about an axis $l$
followed by translation parallel to $l$ by $h\omega t/2\pi$, where $h$ is the {\em pitch} of the screw.}
\label{screw}
\end{center}
\end{figure} 


 
 
 \end{document}
 
 
 
  
 \vspace{.1in}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=1.5in]{skrew.eps}
\caption{\small  A screw motion in $\mathbb{R}^3$ consists of a rotation by $\omega t$ about an axis $l$
followed by translation parallel to $l$ by $h\omega t/2\pi$, where $h$ is the {\em pitch} of the screw.}
\label{screw}
\end{center}
\end{figure} 

