This Problem Sheet is based on Lectures 48, 49, and 50.

• Problem 1 is shows that isometries are volume-preserving, but that the converse can fail.
• Problem 2 is the Divergence Theorem for manifolds with boundary (compare this to Theorem 48.10).
• Problem 3 introduces the curl operator on an oriented Riemannian 3-manifold, and asks you to prove the familiar formula $\operatorname{div} \circ \operatorname{curl} =0$.
• Problem 4 shows that Einstein metrics only become interesting in dimensions at least 4.
• Problem 5 computes the geodesic flow in terms of Jacobi fields.
• Problem 6 is about higher order covariant derivatives.