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?
Olivier has pointed out in the comment that the dot product the OP wrote down is not invariant under change of coordinates and has thus answered the question.