Locally Euclidean Spaces

Abstract

We shall consider a classical Euclidean space in which the points M are marked by a system of general coordinates y¹ where i = 1,2,...n, n being the number of dimensions. This can be any number, but for definiteness we may think of it as equal to 3. In this space a point M’, infinitesimally close to M, has coordinates differing from those of M by dyⁱ and is separated from M by a distance ds = |M’ - M|, given by the metric element

$$ds^2 = g_{ij} dy^i dy^{j^1 } $$

((7.1))

¹ where the set of functions g_ij are functions of the coordinates y^k. The functions g_ij = g_ji make a symmetric tensor, the metric tensor. As we shall see, it is this tensor that determines the type of coordinates used and the space in which they are embedded.