Operational Spacetime pp 29-80 | Cite as

# 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.

### Keywords

Manifold Transportation Rubber Dinates Bilin## Preview

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