Let $M\subset \mathbb{R}^3$ be a connected, compact and orientable surface with positive gaussian curvature. Then, $M$ is diffeomorphic to $\mathbb{S^2}$.
It follows directly from Gauss-Bonnect Theorem that any surface satisfying these conditions is necessarily homeomorphic to a sphere. But how one can prove that, with those same conditions, there's a diffeomorphism between them ?
Thank's for any hint.