Intrinsic Riemannian Geometry

Since many analytic geometric quantities are intrinsic to a smooth m-dimensional surface S in Rⁿ, the standard treatment avoids all reference to an ambient Rⁿ.The surface S is defined as a topological manifold covered by compatible C^∞ coordinate charts, with a "Riemannian metric" ^g (any smooth positive definite matrix). This is not really a more general setting, since J. Nash [Nas] has proved that every such abstract Riemannian manifold can be isometrically embedded in some Rⁿ. suppose that it is a more natural setting, but the formulas get much more complicated.