Riemannian Geometry

Abstract

In Chaps. 3–8 the theory of manifolds was based on the smooth differentiable structure, a complete atlas compatible with a pseudogroup $S$ of class $C^\infty$ introduced at the beginning of Sect. 3.1. With the only exception of Hodge’s star operator introduced at the end of Sect. 5.1, a metric was not needed on general manifolds and on bundles and was not introduced. Since the notion of manifold $M$ was restricted to local homeomorphy with ${\mathbb R}^m$ in this text, by differentiation of real functions along paths in $M$ the tangent space was defined in Sect. 3.3 as a local linearization of $M$. On this basis, tensor bundles, the tensor calculus and the exterior calculus as well as integration of exterior forms could be introduced and the whole theory up to here could be built without a metric on $M$. Now, by defining a metric of a norm on the tangent spaces, due to the locally linear relation between $M$ and its tangent spaces, the manifold $M$ itself is provided with a Riemannian metric. A connection compatible with this metric makes $M$ into a Riemannian geometric space provided with a Riemannian geometry.