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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.

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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.

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    $\begingroup$ It's probably worth noting that this fact opens up what has apparently been a productive area of research into the topological properties of not-a-priori Riemannian manifolds by endowing them with useful metrics; i.e. with respect to the Poincare conjecture. $\endgroup$ – Ryan Reich Aug 31 '13 at 23:10

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