Questions tagged [riemannian-geometry]

Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", i.e. continuous metric tensors.

Riemannian Manifolds have curvatures which can completely be described by a Riemann Curvature Tensor, which is given by:

$$R_{\mu\nu\rho}^\sigma=\mathrm{d}x^\sigma[\nabla_\mu,\nabla_\nu]\partial_\sigma$$

A partial trace of this tensor is a symmetric tensor, namely, the Ricci Curvature Tensor $R_{\mu\nu}=g^{\rho\sigma}R_{\mu\nu\rho\sigma}$, which is very useful in General Relativity, for example. In 4-dimensions, the Riemann Curvature Tensor can completely be described by the Ricci Curvature Tensor and the Weyl Tensor $C_{\mu\nu\rho\sigma}$.

The Riemann Curvature Tensor also satisfies a number of identities called the *Bianchi Identities".