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 beyond $\mathbb{R}^n$. General relativity is written in this language.
Riemannian Geometry
An important subdiscipline of differential geometry is Riemannian geometry, which introduces a metric-tensor 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 a 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 general-relativity is the most famous example of applications of differential geometry to physics, there are many others. electromagnetism and gauge-theory can be formulated in the language of fiber bundles, which are particular examples of manifolds. The hamiltonian-formalism or classical-mechanics can be formulated in terms of symplectic manifolds.
