A quantitative approach to the regularity of a Riemannian surface

We introduce two definitions with the purpose of quantifying the concept of a $C^{2,\alpha }$ surface for $0<\alpha <1$. The intrinsic definition is given in terms of the $\alpha$-Hölder norm of the Gauss curvature function. The extrinsic one relies on the existence of a smooth local representation of the Riemannian metric. We show that these definitions are equivalent up to a constant depending on $\alpha$.