Differential Geometry of Curves and Surfaces, 2nd Edition

CHAPTER 6

The First and Second Fundamental Forms

Recall that the local geometry of space curves is completely determined by two geometric invariants: the curvature and the torsion. Similarly, as we shall see, the local geometry of a regular surface S in ℝ³ is determined by the first and second fundamental forms.

The value of restricting attention to regular surfaces is that at all points on a regular surface, there is an open neighborhood regularly homeomorphic to ℝ² via a parametrization X→. Thus, at a point p ∈ S, with p=X→(q), the differential dX→q provides a natural isomorphism between ℝ² and T_pS. Whenever we consider vectors on S based at the point p, we must consider them as elements of T_pS, and we can “do geometry” locally on S by identifying ...