Riemannian Manifolds

Abstract

In this chapter, general structures of differential manifolds are given, especially the operational aspects of Riemannian manifolds. Manifolds as such are rather amorphous structures. Physically important and manifold-characterizing are the operation groups, they are parametrizing. A Riemannian manifold has a global and a local invariance group in addition to its tangent Poincaré or Euclidean group; e.g., for the sphere $\Omega^2$ and the hyperboloid y ² given by the rotations SO(3) and Lorentz transformations SO ₀(1, 2), respectively, as the global groups (motion groups), and, for both, the axial rotations SO(2) as the local group and the Euclidean group ${\bf {SO}}\left( 2 \right)\vec \times \mathcal{R} ^2$ as the tangent group. The Einstein cosmos $\mathcal{R} \times \Omega^3$ has $\mathcal{R} \times {\bf {SO}}\left( 4 \right)$ as the global group, SO(3) as the local group, and ${\bf {SO}}\left( 1,3 \right)\vec \times \mathcal{R} ^4$ as the tangent Poincaré group.

After a discussion of Riemannian manifolds with maximal symmetries and constant curvature, i.e., spheres, flat spaces and timelike hyperboloids, the relationship between coset spaces of real simple Lie groups and manifolds with a covariantly constant curvature as classified by Cartan is presented.