Intrinsic Riemannian Geometry
DOI link for Intrinsic Riemannian Geometry
Intrinsic Riemannian Geometry book
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.