Intrinsic Geometry of Surfaces: Local Theory

Abstract

We are now going to concentrate on the properties of a surface ƒ: U→ℝ3 which are intrinsic in the sense that they are definable in terms of tangent vectors to the surface and the first fundamental form and its derivatives. For example, the length of a vector or the length of a curve on a surface are intrinsic quantities. The Gauss curvature and the curvature tensor are also intrinsic since they may be defined in terms of the first fundamental form and its derivatives. In contrast, the second fundamental form is not intrinsic. It requires discussion of normal vector fields and cannot, in any case, be reduced to the first fundamental form. Also, principal curvatures are not intrinsic, even though their product, the Gauss curvature, is an intrinsic quantity.