# 53. The Hopf-Rinow Theorem and its friends

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:

- 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 $\mathbb{R}^n$. - The
*Bonnet-Myers Theorem*:

##### 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?