TITLE: Compatibility of a connection and metric
QUESTION [6 upvotes]: Every Riemannian manifold admits a metric connection. Suppose $M
$ is a manifold and $\nabla $ is an arbitrary connection on the tangent bundle. Does $M$ necessarily admit a metric such that $\nabla$ is compatible with this metric?

REPLY [3 votes]: Take a look at https://mathoverflow.net/questions/54434/when-can-a-connection-induce-a-riemannian-metric-for-which-it-is-the-levi-civita/, especially Thurston's answer, although it doesn't answer your question, but a more special situation.