We know that a Riemannian metric on a smooth manifold $M$ is a smoothly varying assignment of an inner product to every tangent space of $M$.
Question 1: Do all inner products on $\Bbb R^n$ are equivalent? i.e. one can deform one to another smoothly?
Sorry for the following because I don't know it make sense or not.
Question 2: Does exotic structure on inner product space $\Bbb R^n$ (if exist) induces an exotic structure on Riemannian manifold $M$ with $\Bbb R^n$ as tangent space?