conformally euclidean metric on riemannian surface

Question

We know that in euclidean space $\mathbb{R}^3$ the euclidean metric induces on a 2-dimensional surface a riemannian metric which can be brought into conformal form by means of a local change of coordinates; can you provide a counterexample to this: for any n-dimensional riemannian manifold we can find a change of coordinates to conformal metric ?

Answer 1

You're asking if every $n$-dimensional Riemannian manifold is conformally flat. As noted in the example section of this Wikipedia page, there are well-understood obstructions to this. In particular, any simply-connected closed $n$-manifold not diffeomorphic to $S^n$ will give you a counter-example (see also this MathOverflow answer by Robert Bryant). Examples of such are exotic spheres.