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\title{{\bf  Homework set 2 - due 09/13/24}\\ {\large Math 5047} 
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 Turn in problems 2(b,c), 3, 4(a,b), 5(a,b).
\begin{enumerate}
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\item {\bf Reading.} Read Chapter 4 of Lee's text, pages 84-103.
\item\label{problem2}   {\bf Problem 4-1, page 111, Lee.} Let $M\subseteq \mathbb{R}^n$ be an embedded submanifold  and $Y\in \mathfrak{X}(M)$.  For every point $p\in M$ and vector $v\in T_pM$, define $\nabla^{}_vY$ by
$$ \nabla^{}_vY=\Pi_p^{} D_v \tilde{Y},$$
where $\tilde{Y}$ is a smooth vector field defined on an open subset of $\mathbb{R}^n$ containing $p$ and $\Pi_p^{}: T_p\mathbb{R}^n\rightarrow T_pM$ is the orthogonal projection (relative to the Euclidean dot-product in $\mathbb{R}^n$).
\begin{enumerate}
\item Show that $\nabla_v^{} Y$ does not depend on the choice of extension $\tilde{Y}$ of $Y$. [Lee offers a hint, which I find confusing, so I won't write it here.]
\item It is already shown in Example 4.9 that $\nabla^{}$ is an affine connection on $TM$. Now show that it is torsion free:
$$ \nabla^{}_XY-\nabla^{} _YX=[X,Y],$$
for $X,Y\in \mathfrak{X}(M)$. (Local extensions of $X, Y$ exist by Exercise A.23.)
\item Show that $\nabla^{}$ is compatible with the induced Riemannian metric on $M$. That is, show that for $X, Y\in \mathfrak{X}(M)$ and $v\in T_pM$,
$$ v(X\cdot Y)=(\nabla_v^{}X)\cdot Y+X\cdot \nabla_v^{} Y$$
where $\cdot$ is the ordinary dot-product on $\mathbb{R}^n$. This makes sense since tangent vectors to $M$ may be regarded as vectors in $\mathbb{R}^n$. 
\item Show that $\nabla^{}$ is invariant under rigid motions of $\mathbb{R}$, in the following sense: if $F$ is an element of the Euclidean group (the group of transformations of $\mathbb{R}^n$ generated by rotations and translations), then 
$$ dF_p \nabla_vY=\nabla_{dF_pv} F_*Y.$$
Here $F$ maps $M$ to $F(M)$ and we use the same notation $\nabla$ for the connection on $TM$ and on $TF(M)$. 
\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\bf Problem 4-6, page 112, Lee.} Let $M$ be a smooth manifold and let $\nabla$ be a connection in $TM$. Define
a map $T:\mathfrak{X}(M)\times\mathfrak{X}(M)\rightarrow \mathfrak{X}(M)$ by
$$T(X,Y)=\nabla_XY-\nabla_YX-[X,Y]. $$
\begin{enumerate}
\item Show that $T$ is a $(1,2)$-tensor field, called the {\em torsion tensor} of $\nabla$. The connection is called {\em symmetric} or {\em torsion free} if $T=0$. For example, the connection of Problem \ref{problem2} is symmetric.
\item Let $\frac{\partial}{\partial x_i}$ be coordinate vector fields on a local coordinate chart of $M$. Define the {\em Christoffel symbols} $\Gamma_{ij}^k$ as the functions given  by
$$ \nabla_{\frac{\partial}{\partial x_i}}\frac{\partial}{\partial x_j}=\sum_k \Gamma_{ij}^k \frac{\partial}{\partial x_k}.$$
Show that $\nabla$ is symmetric if and only if on every coordinate chart, $\Gamma_{ij}^k=\Gamma_{ji}^k$ for all $i, j, k$.
\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\bf Problem 4-11, page 113, Lee, expanded.} Suppose $G$ is a connected Lie group. 
\begin{enumerate}
\item Show  that there is a unique connection $\nabla$ on $TG$ with the property that every left-invariant vector field is parallel. (Note: a vector field $X$ is parallel relative to an affine connection $\nabla$ if $\nabla X=0$.)
\item Show that   $\nabla$ is torsion free if and only if $G$ is abelian. (For this assignment, take for granted that if the Lie algebra is abelian then the connected Lie group is abelian. We'll discuss this further in class.)
\item Define $\tau$ to be the $(1,2)$-tensor field on $G$ such that on $u,v\in T_gG$,
$$ \tau_g(u,v)=[X_u,X_v]_g,$$
where $X_u$ is the unique left-invariant vector field on $G$ equal to $u$ at $g$.  (Convince yourself that $\tau$ is, indeed, a tensor field and that
$\tau(X,Y)=[X,Y]$ on left-invariant vector fields.)
Now define a connection on $G$ as follows: 
$$ \overline{\nabla}_XY=\nabla_XY +\frac12\tau(X,Y).$$
Thus if $X, Y$ are left-invariant, $\overline{\nabla}_XY=\frac12[X,Y]$.
Show that $\overline{\nabla}$ is still a connection on $TM$ and it is  torsion free. (Even if $G$ is non-abelian.)
\item Show that both $\nabla$ and $\overline{\nabla}$ are invariant under left-translations. That is, show the following: let $F=L_g:G\rightarrow G$ be
the diffeomorphism of $G$ such that $F(h)=gh$ for all $g, h\in G$. This is the {\em left translation map}. Then
$$ F_*\nabla_XY=\nabla_{F_*X}F_*Y.$$
\item Show that both $\nabla$ and $\overline{\nabla}$ are also invariant under right translations: if $F=R_g:G\rightarrow G$, $F(h)=hg$, then 
$$ F_*\nabla_XY=\nabla_{F_*X}F_*Y.$$
For this, note that if $X$ is a left-invariant vector field, then $(R_g)_*X$ is also left-invariant. (This is because $R_{g}$ and $L_{h}$ commute for all $g, h\in G$, something easy to check.)
\end{enumerate}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\bf Problem 4-14, page 113, Lee.} Let $M$ be a smooth $n$-manifold and $\nabla$ a connection in $TM$, let $(E_1, \dots, E_n)$ be a local frame
on some open subset $U\subseteq M$, and let $(\epsilon_1, \dots, \epsilon_n)$ be the dual coframe. 
\begin{enumerate}
\item Show that there is a uniquely determined $n\times n$ matrix of smooth $1$-forms $\left(\omega_{ij}\right)$ on $U$, called the {\em connection $1$-forms}
for this frame, such that 
$$ \nabla_X E_i = \sum_j \omega_{ij}(X)E_j$$
for all $X\in \mathfrak{X}(M)$.
\item {\em Cartan's first structure equation.} Prove that these forms satisfy the following equation, due to \'Elie Cartan:
$$d\epsilon_j = \sum_{i=1}^n \epsilon_i \wedge \omega_{ij} +\tau_j, $$
where $\tau_1, \dots, \tau_n\in \Omega^2(M)$ are the {\em torsion $2$-forms}, defined in terms of the torsion tensor and the local frame by:
$$T(X, Y) =\sum_{i=1}^n\tau_i(X, Y)E_i. $$
\end{enumerate}
   
 

 

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