This Problem Sheet covers Lectures 42, 43, and 44.

• 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 I skipped in class.
• Problem 4 studies bi-invariant Riemannian metrics on Lie groups.
• Problem 5 identifies the Levi-Civita connection on a Lie group endowed with a bi-invariant Riemannian metric.