Do all parallelizable manifolds admit a flat metric?

Question

A simply connected manifold $M$ admits a flat metric/connection if and only if $M$ is parallelizable.

What happens/can happen if $M$ is notsimply-connected?

Definitions:

https://en.wikipedia.org/wiki/Parallelizable_manifold

https://en.wikipedia.org/wiki/Simply_connected_space

Answer 1

Your assertion is not true, the 3-sphere $S^3$ is a Lie group so it is parallelizable but it does not admits a flat metric, since compact manifolds which admit a flat metric are finitely covered by the n-torus $T^n$.

Answer 2

Each parallelizable $n$-dimensional manifold $M$ admits a flat affine connection on $TM$ (the trivial connection). This, of course, does not yield a flat metric. However, if you have a connected noncompact parallelizable manifold (no need to assume simply connected), it does admit a (typically incomplete) flat Riemannian metric. This is an application of the Hirsch-Smale immersion theory: Find an immersion $M\to R^n$ and pull-back the flat metric. On another hand, if $M$ is simply connected and $TM$ has a flat affine connection, then $M$ is indeed parallelizable. (If the connection is complete, then even more is true: $M$ is diffeomorphic to $R^n$.)