• Problem 1 is a fun™ computation in local coordinates.
• Problem 2 gives another viewpoint on the torsion tensor of a connection (compare this to the Bianchi Identity $d^{ \nabla^{ \operatorname{Hom}}} (R^{ \nabla} ) = 0$.)
• Problem 3 asks you to prove the second additional curvature symmetry of torsion-free connections that we skipped in class.
• Problem 4 identifies the Levi-Civita connection of the round metric on $S^m$.
• Problem 5 introduces Killing fields on a Riemannian manifold $(M,g)$, which are vector fields $X$ such that $\mathcal{L}_X g= 0$.
• Problem 6 asks you to show that if $\varphi \colon M \to N$ is an isometric map then for $\xi \in T_{ \varphi(p)}N$, the tangential component $\xi^\top$ is simply the orthogonal projection of $\xi$ onto $D \varphi(p)(T_pM)$.
• Problem 7 is about Riemannian coverings, and requires you to know a little bit of covering space theory in order to solve it.
• Problem 8 is about invariant metrics on homogeneous spaces.
• Problem 9 completes our discussion of holonomy groups.
• Problem 10 studies bi-invariant Riemannian metrics on Lie groups.
• Problem 11 identifies the Levi-Civita connection of a Lie group endowed with a bi-invariant Riemannian metric.