A Brief Guide to the Local Theory of Surfaces in R³

Abstract

The local study of curves in R², in sections 4 and 5 of chapter 9, was simple: theorem 8.5.7 shows that a single invariant, the curvature (expressed as a function of the arclength), is enough to characterize such a curve. The fundamental reason for this simplicity, and one that remains true no matter what the dimension of the ambient space, is that the intrinsic geometry of curves is trivial; the metric given by the length of paths on a curve is always the same as the metric on some interval of R. And the “shape”, or “position,” of a curve in R² is specified by a mere scalar function, the curvature.