# Problem Sheet W

This Problem Sheet is based on Lectures 51, 52, and 53.

- Problem 1 shows that Proposition 53.10 is not an “if and only if” statement—a geodesic can still fail to be minimal even without conjugate points.
- Problem 2 shows that a isometric map between Riemannian manifolds of the same dimension is automatically a Riemannian covering if the domain is complete.
- Problem 3 relates conjugate points with the rank of the exponential map.
- Problem 4 asks you to carry out the computation of the Hessian of the energy functional that I skipped in class.
- Problem 5 shows how curvature bounds can allow one to infer information about conjugate points.

**But wait, there’s more! **A “bonus” problem sheet is also available here.

Comments and questions?