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\title{Smooth Manifolds Fall 2022 - Homework 8}

\begin{document}

\maketitle

\begin{problem}
Find an area form $\omega$ on $S^2$ such that for any rotation about some axis in $\R^3$, $R : S^2 \to S^2$, we have that $R^* \omega = \omega$. Prove the invariance property, and show that this volume form is unique up to positive scalar multiple.  

[{\it Remarks}: You may use the natural coordinates for the tangent spaces as subspaces of $\R^3$ (ie, you may write down a form on $\R^3$ and restrict it to $S^2$ to construct the form). You may use the fact that rotations around the coordinate axes generate the group of all rotations.]
\end{problem}


\begin{problem}\hspace{1cm}

\begin{enumerate}
\item Prove the following convenient formula for the exterior derivative of a 1-form $\alpha$, where $X$ and $Y$ are vector fields on $M$. Note that if $f \in C^\infty(M)$, then $X \cdot f$ denotes the derivative of $f$ along the vector field $X$.

\[ d\alpha(X,Y) = X \cdot (\alpha(Y)) - Y \cdot (\alpha(X)) - \alpha([X,Y]) \] 
 [{\it Hint}: Any 1-form is locally a linear combination of forms of the form $u \, dv$ for $u,v \in C^\infty$. Evaluate both side of the desired equality for such forms.]
\item Use this to find a formula for $\mc L_X\alpha(Y)$, where $X$ and $Y$ are $C^\infty$ vector fields and $\alpha$ is a 1-form on $M$. Think magically!
\end{enumerate}
\end{problem}

\begin{problem}
Fix a $2n$-dimensional manifold $M$. Recall that a 2-form $\omega$ is called {\it symplectic} if $d\omega = 0$ and the $n$-fold wedge product of $\omega$ is a volume form on $M$. 

\begin{enumerate}
\item  Show that a closed 2-form $\omega$ is symplectic if and only if for every $x\in M$ and nonzero vector $X \in T_xM$, there exists $Y \in T_xM$ such that $\omega(X,Y)\not= 0$. [{\it Hint}: One direction is an easy calculation by contradiction using the operator $\iota_X$ and its formula for wedge products. For the other direction, assume that you have found linearly independent vectors $X_1,Y_1,\dots,X_k,Y_k \in T_xM$ such that $\omega(X_i,Y_i) = 1$ for every $i$, and all other pairwise combinations evaluate to 0. Define:

\[ \phi_x^{(k)} : T_xM \to T_xM \qquad \phi_x^{(k)}(v) = v -\left( \sum_{i=1}^k \omega(v,X_i)Y_i - \omega(v,Y_i)X_i\right)\]

Choose $\tilde{X}_{k+1}, \tilde{Y}_{k+1}$ to be a pair such that $\tilde{X}_{k+1}$ is not in the span of the $\set{X_i,Y_i}_{i=1}^k$ and $\omega(\tilde{X}_{k+1},\tilde{Y}_{k+1}) = 1$, and let $X_{k+1} = \phi_x^{(k)}(\tilde{X}_{k+1})$, $Y_{k+1} = \phi_x^{(k)}(\tilde{Y}_{k+1})$. Show that $X_{k+1}$ and $Y_{k+1}$ extend the desired properties.]
\item Show that if $\alpha$ is any 1-form on $M$, there exists a unique vector field $X_\alpha$ such that $\iota_{X_\alpha}\omega = \alpha$. [{\it Hint}: You don't have to show regularity, and you can build the vector field pointwise by showing that the map $F_x : T_xM \to T_x^*M$ defined by $F_x(v) = \iota_v\omega(x)$ is an isomorphism of vector spaces]
\end{enumerate}
\end{problem}


\end{document}




