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Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.

Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $$\mathbb{R}^n$$. General relativity is written in this language.

### Riemmanian Geometry

An important subdiscipline of differential geometry is Riemannian geometry, which introduces a to measure geometric properties on the space, such as angles between vectors, lengths, and so on. In a Riemannian manifold, all lengths are positive (or vanishing, for coincident points). In a pseudo-Riemannian manifold, some curves might have negative length. General relativity uses pseudo-Riemannian manifold, and the single negative direction is interpreted as time.

Riemannian Manifolds have curvatures which can completely be described by a Riemann curvature tensor, which is given by the tensor $$R_{\mu\nu\rho}^\sigma=\mathrm{d}x^\sigma[\nabla_\mu,\nabla_\nu]\partial_\sigma.$$ 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".

### Applications

While is the most famous example of application of differential geometry to physics, there are many others. and can be formulated in the language of fiber bundles, which are particular examples of manifolds. The or can be formulated in terms of symplectic manifolds.