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    \begin{document}
    \begin{center}
    \textbf{ MATH534A, Problem Set 5, due Oct 9}
    \end{center}
    All problems are worth the same number of points.
    \begin{enumerate}
    \item Prove that the set of points in $\R^2$ satisfying the equation $xy = 0$ is not a smooth submanifold of $\R^2$.
    \item Let $Z$ be a smooth manifold, $Y$ be a smooth submanifold of $Z$, and $X$ be a smooth submanifold of $Y$. Show that $X$ is a smooth submanifold of $Z$. In other words, being a submanifold is a transitive relation.
    \item \begin{enumerate} \item Show that the manifold $\SO_n(\R)$, consisting of $n \times n$ real orthogonal matrices with determinant~$1$, is connected.
    \item  Show that the manifold $\O_n(\R) \setminus \SO_n(\R)$, consisting of $n \times n$ real orthogonal matrices with determinant~$-1$, is diffeomorphic to $\SO_n(\R)$ and hence also connected.
    \end{enumerate}
    \item Show that $\U_n$, the set of $n \times n$ unitary matrices, is a smooth submanifold of $\Mat_{n \times n}(\C) \simeq \R^{2n^2}$ and find its dimension.
\item Prove that $\SO_2(\R)$ is diffeomorphic to $S^1$.
    \end{enumerate}
    \end{document}