Non Riemannian Manifolds and Failure of the Usual Dot Product to induce a Riemannian Metric

Let $M$ denote a smooth manifold. Let $X = X^i\frac{\partial}{\partial x^i}$ and $Y=Y^i\frac{\partial}{\partial x^i}$ be smooth vector fields on $M$. We can pointwise define an inner product via the usual dot product. This gives us a smooth map $p \rightarrow \Sigma X^i(p)Y^i(p)$ for $p \in M$.

To me, this seems to imply that the usual dot product always induces a Riemannian metric, contradicting the existence of non-Riemannian manifolds.

Can someone point out the error in the above reasoning?

• This inner product depends on the chart, and will not be defined globally. What do you mean by "non-Riemannian manifold" ? Vector bundles over paracompact base spaces always carry riemannian structures. – Olivier Bégassat May 20 '17 at 18:20
• I'm not sure where the above reasoning is dependent on paracompactness. – graviton May 20 '17 at 18:29