In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced a way to describe it as a "little monster tensor". Similar notions have found applications everywhere in differential geometry. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... Bernhard Riemann. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ...

For a more elementary discussion see the article on curvature which discusses the curvature of curves and surfaces in 2 and 3 dimensions. Curvature refers to a number of loosely related concepts in different areas of geometry. ...

The curvature of a Pseudo-Riemannian manifold can be expressed in the same way with only slight modifications. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ...

Ways to express the curvature of a Riemannian manifold

The curvature tensor

The curvature of 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) and Lie bracket [ * , * ] by the following formula: In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor fields over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented...

Here R(u,v) is a linear transformation of the tangent space of the manifold; it is linear in each argument. If and are coordinate vector fields then [u,v] = 0 and therefore the formula simplifies to

i.e. the curvature tensor measures noncommutativity of the covariant derivative.

The linear transformation is also called the curvature transformation or endomorphism.

NB. There are a few books where the curvature tensor is defined with opposite sign.

If you change the metric by a factor e^2f, the curvature tensor changes to (seen as a (0,4)-Tensor):

where denotes the Kulkarni-Nomizu product. The Kulkarni-Nomizu product is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor. ...

Symmetries and identities

The curvature tensor has the following symmetries:

The last identity was discovered by Ricci, but is often called the first Bianchi identity, just because it looks similar to the Bianchi identity below. These three identities form a complete list of symmetries of the curvature tensor, i.e. given any tensor which satisfies the identities above, one could find a Riemannian manifold with such a curvature tensor at some point. Simple calculations show that such a tensor has n²(n² − 1) / 12 independent components. Yet another useful identity follows from these three: Gregorio Ricci-Curbastro (January 12, 1853 - August 6, 1925) was an Italian mathematician. ...

The Bianchi identity (often the second Bianchi identity) involves the covariant derivatives:

Sectional curvature

Sectional curvature is a further, equivalent but more geometrical, description of the curvature of Riemannian manifolds. It is a function K(σ) which depends on a section σ (i.e. a 2-plane in the tangent spaces). It is the Gauss curvature of the σ-section at p; here σ-section is a locally-defined piece of surface which has the plane σ as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of σ under the exponential map at p. In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...

If v,u are two linearly independent vectors in σ then

The following complex formula indicates that sectional curvature describes the curvature tensor completely:

Curvature form

The Cartan formalism gives a very elegant way to describe curvature. It is used more for general vector bundles, and for principal bundles, but it works just as well for the tangent bundle with the Levi-Civita connection. The curvature of n-dimensional Riemannian manifold is given by an antisymmetric n×n matrix of 2-forms (or equivalently a 2-form with values in so(n), the Lie algebra of the orthogonal group O(n), which is the structure group of the tangent bundle of a Riemannian manifold). In differential geometry, the curvature form describes curvature of principal bundle with connection. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji   for all i and j. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ...

Let e_i be a local section of orthonormal bases. Then one can define the connection form, an antisymmetric matrix of 1-forms which satisfy from the following identity

Then the curvature form is defined by

The following describes relation between curvature form and curvature tensor:

This approach builds in all symmetries of curvature tensor except the first Bianchi identity, which takes form

where θ = θⁱ is an n-vector of 1-forms defined by . The second Bianchi identity takes form

DΩ = 0

D denotes the exterior covariant derivative In differential geometry, the connection form describes connection on principal bundles (or vector bundles). ...

The curvature operator

It is sometimes convenient to think about curvature as an operator Q on tangent bivectors (elements of Λ²(T)), which is uniquely defined by the following identity: In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ... Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product. ...

It is possible to do this precisely because of the symmetries of the curvature tensor (namely antisymmetry in the first and last pairs of indices, and block-symmetry of those pairs).

Further curvature tensors

In general the following tensors and functions do not describe the curvature tensor completely, however they play an important role.

Scalar curvature

Scalar curvature is a function on any Riemannian manifold, usually denoted by Sc. It is the full trace of the curvature tensor; given an orthonormal basis {e_i} in the tangent space at p we have In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

where Ric denotes Ricci tensor. The result does not depend on the choice of orthonormal basis. Starting with dimension 3, scalar curvature does not describe the curvature tensor completely. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

Ricci curvature

Ricci curvature is a linear operator on tangent space at a point, usually denoted by Ric. Given an orthonormal basis {e_i} in the tangent space at p we have In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

The result does not depend on the choice of orthonormal basis. Starting with dimension 4, Ricci curvature does not describe the curvature tensor completely.

Explicit expressions for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). ... In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. ...

Weyl curvature tensor

The Weyl curvature tensor has the same symmetries as the curvature tensor, plus one extra: its Ricci curvature must vanish. In dimensions 2 and 3 Weyl curvature vanishes, but if the dimension n > 3 then the second part can be non-zero. In differential geometry, the Weyl curvature tensor is the traceless component of the Riemann curvature tensor. ...

• The curvature tensor can be decomposed into the part which depends on the Ricci curvature, and the Weyl tensor.
• If g′=fg for some positive scalar function f — a conformal change of metric — then W ′ = W.
• For a manifold of constant curvature, the Weyl tensor is zero.
  • Moreover, W=0 if and only if the metric is locally conformal to the standard Euclidean metric (equal to fg, where g is the standard metric in some coordinate frame and f is some scalar function).

In mathematics, a conformal map is a function which preserves angles. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. ... In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ...

Calculation of curvature

For calculation of curvature

• of hypersurfaces and submanifolds see second fundamental form,
• in coordinates see the list of formulas in Riemannian geometry or covariant derivative,
• by moving frames see Cartan connection and curvature form.
• the Jacobi equation can help if one knows something about the behavior of geodesics.