Is there a Smooth Real Manifold which is not a Riemannian Manifold?

Question

I am taking a course in Differential Topology right now, but I know of another Subject called "Riemannian Geometry" which studies Riemannian Manifolds. The definition of a real smooth manifold and a Riemannian Manifold are very similar except for the detail that makes them Riemannian (having a inner product on the tangent space of every element which varies smoothly with respect to this element of the R. Manifold).

So my question is: is there any example of a Smooth Manifold that is NOT a Riemannian Manifold?

Thanks.

Answer 1

Every paracompact smooth manifold can be equipped with a Riemannian metric. To see this note that $\mathbb{R}^n$ has a Riemannian metric so you can construct a Riemannian metric on any smooth manifold by using a partition of unity subordinate to a locally finite atlas of charts.