Let $(M,g)$ be a smooth compact Riemannian manifold of dimension at least 3. Does the Hopf-Rinow theorem imply that $(M,g)$ is a complete Riemannian manifold? I.e. is a compact Riemannian manifold always complete?
If so, then can we conclude from the Cartan-Hadamard theorem that if $(M,g)$ is a compact and simply connected Riemannian manifold, then it must have positive sectional curvature?
If so, this seems to contradict the answer to this question, which says that any Riemannian manifold of dimension at least 3 admits a complete Ricci negative metric.
Where is the error in my reasoning?
What is an example of a sphere with negative sectional or Ricci curvature?