Spherical and Hyperbolic Spaces

Abstract

Among the jungle of Riemannian spaces, we have selected out the remarkable class with constant curvature: locally non-Euclidean spaces which still are of extraordinary variety, as we have seen at the end of the previous chapter. Among locally Euclidean spaces, showing also a large variety of forms, the one which is simply-connected, the Euclidean space, appears as the simplest example. Similarly here, among locally non-Euclidean spaces those which are simply-connected will also have the barest essentials, and these are the easiest to study. This chapter is devoted to two of the simply-connected locally non-Euclidean spaces, simply called non-Euclidean spaces. One, with positive curvature K, is the spherical space, and the other, with negative curvature K, is the hyperbolic space. We shall not speak again of space with zero curvature, Euclidean space.