Differential Geometry of Curves and Surfaces, 2nd Edition

CHAPTER 7

The Fundamental Equations of Surfaces

In the previous chapter, we began to study the local geometry of a surface and saw how many computations depend on the coefficients of the first and second fundamental forms. In fact, in Section 6.1 we saw that many metric calculations (angles between curves, arc length, area, etc.) depend only on the first fundamental form. We first approached such concepts from the perspective of objects in ℝ³, but with the first fundamental form, one can perform all the calculations by using only the coordinates that parametrize the surface, without referring to the ambient space. In fact, it is not at all difficult to imagine a regular surface as a subset of ℝⁿ, with n > 3, and the formulas that depend on ...