In
mathematics, specifically
differential geometry, the
infinitesimal geometry of
Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.
Riemann
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
introduced an abstract and rigorous way to define curvature for these manifolds, now known as the
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
. Similar notions have found applications everywhere in differential geometry.
For a more elementary discussion see the article on
curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions, as well as the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
.
The curvature of a
pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
can be expressed in the same way with only slight modifications.
Ways to express the curvature of a Riemannian manifold
The Riemann curvature tensor
The curvature of a Riemannian manifold can be described in various ways; the most standard one is the curvature tensor, given in terms of a
Levi-Civita connection (or
covariant differentiation
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differ ...
)
and
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...