Math 6510 • Differentiable Manifolds

\begin{enumerate}
\item[(a)] What is a subalgebra of a Lie algebra?

\item[(b)] Let \(G\) be a Lie group, with Lie algebra \(\mathfrak{g}\). Prove that
any Lie subgroup \(H\) of \(G\) corresponds to a subalgebra of \(\mathfrak{g}\).

\item[(c)] What is a smooth plane field on \(G\)?

\item[(d)] Give an example of a foliation on a 2-torus that has compact leaves
and noncompact leaves.

\item[(e)] State Frobenius' theorem.

\item[(f)] Prove that any subalgebra of \(\mathfrak{g}\) corresponds to a
foliation of \(G\).

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\item[(g)] Given a subalgebra of \(\mathfrak{g}\), and the corresponding foliation
of \(G\), prove that the leaf containing the identity is a group, and thus that
there is a correspondence between connected Lie subgroups of \(G\) and subalgebras
of \(\mathfrak{g}\).
\end{enumerate}