We begin this final lecture by proving the Hopf-Rinow Theorem, which states that a Riemannian metric $m$ on a smooth manifold $M$ is complete (i.e. all geodesics defined for all times) if and only if $(M,d_m)$ is complete as a metric space (i.e. all Cauchy sequences converge).

We then prove that a non-constant geodesic is never length-minimising after the first conjugate point. Combining this result with known behaviour for spaces of constant curvature, we obtain two more results:

  1. The Cartan-Hadamard Theorem: if $(M^n,m)$ is a complete Riemannian manifold with non-positive sectional curvature then the universal cover $ \tilde{M}$ of $M$ is diffeomorphic to $\mathbb{R}^n$.
  2. The Bonnet-Myers Theorem: if $(M^n,m)$ is a complete Riemannian manifold with with uniformly positive sectional curvature then both the diameter and the fundamental group of $M$ are finite.
Remarks
  • The last Problem Sheet W is now available. (Oh, and there is also a bonus Problem Sheet…)
  • A complete pdf of Lectures 1–53, and all the problem sheets and their solutions will appear shortly on my forum. (I will also update the $( \clubsuit)$ marks to clarify exactly what is and is not examinable).
  • 🙃 Thank you everyone for suffering through my course this year! 🙃 I hope you all enjoy your summer vacation studying for your exams.

Comments and questions?