ABSTRACT
Riemannian metrics, curvature, and the associated theorems for geodesics gave Einstein precisely the mathematical tools he needed to express his conception of a curvilinear spacetime. Applications of manifolds to either geometry or physics may require a different structure from or additional structure to a Riemann metric. Turning to physics, general relativity, one of the landmark achievements in science of the early 20th century, stands as the most visible application of Riemann manifolds to science. Many examples of Riemannian metrics arise naturally as submanifolds of Riemannian manifolds. Using integration, the Riemannian metric allows for formulas that measure nonlocal properties, such as length of a curve and volume of a region on a manifold. In the context of Riemannian geometry, it is natural to wish for a connection that is in some sense “nice” with respect to the metric on the manifold.
