Riemannian manifold

A torus naturally carries a Euclidean metric, obtained by identifying opposite sides of a square (left). The resulting Riemannian manifold, called a flat torus, cannot be isometrically embedded in 3-dimensional Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the ${\displaystyle n}$-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all Riemannian manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them.

Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.

Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the ambient space. The same is true for any submanifold of Euclidean space of any dimension. Although John Nash proved that every Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract space itself without referencing an ambient space. In many instances, such as for hyperbolic space and projective space, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics on Lie groups and homogeneous spaces are defined intrinsically by using group actions to transport an inner product on a single tangent space to the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using tools from partial differential equations.

Riemannian geometry, the study of Riemannian manifolds, has deep connections to other areas of math, including geometric topology, complex geometry, and algebraic geometry. Applications include physics (especially general relativity and gauge theory), computer graphics, machine learning, and cartography. Generalizations of Riemannian manifolds include pseudo-Riemannian manifolds, Finsler manifolds, and sub-Riemannian manifolds.