The Notion of a Riemannian Metric

Abstract

In the preceding chapter we associated with each curvilinear coordinate system x = (x ¹, ... , x ⁿ) on a domain C a smooth functional matrix G(x), and these matrices, at each point, behave as a quadratic form under a change-of-coordinate transformations. The matrices G(x) arose from the formula for the element of length of a curve, and we saw that the length of a curve can be calculated in a curvilinear coordinate system using the G(x). The property of transforming at each point like a quadratic form turns out to be of major importance.