Questions tagged [differential-geometry]

Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

An important subdiscipline of differential geometry is 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".