Say we have a smooth manifold $M$. Does there always exist a metric function $d : M \times M \to \mathbb{R}$ such that $d$ is smooth and gives rise to the same topology of the manifold $M$ ?
I am inclined to think that the answer is affirmative. One could argue that by strong Whitney embedding theorem , $M$ could be embedded in the Euclidean space. Now the restriction of the ordinary metric on the Euclidean space to the manifold would satisfy the conditions. Am I right, or am I making some mistake ? Thanks !