Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifoldnegatively curvedRiemann curvaturesectional curvature
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.wikipedia
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Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
Riemann 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.

Curvature

curvednegative curvatureextrinsic curvature
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. It is the Gauss curvature of the \sigma-section at p; here \sigma-section is a locally defined piece of surface which has the plane \sigma as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of \sigma under the exponential map at p.
See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor.

Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometry
It is the Gauss curvature of the \sigma-section at p; here \sigma-section is a locally defined piece of surface which has the plane \sigma as a tangent plane at p, obtained from geodesics which start at p in the directions of the image of \sigma under the exponential map at p.
The exponential map is also useful in relating the abstract definition of curvature to the more concrete realization of it originally conceived by Riemann himself—the sectional curvature is intuitively defined as the Gaussian curvature of some surface (i.e., a slicing of the manifold by a 2-dimensional submanifold) through the point p in consideration.

Manifold

manifoldsboundarymanifold with boundary
Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection.

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
Explicit expressions for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols.

Kulkarni–Nomizu product

Kulkarni-Nomizu product
where denotes the Kulkarni–Nomizu product and Hess is the Hessian.
For this very reason, it is commonly used to express the contribution that the Ricci curvature (or rather, the Schouten tensor) and the Weyl tensor each makes to the curvature of a Riemannian manifold.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
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) \nabla and Lie bracket by the following formula:

Mathematics

mathematicalmathmathematician
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.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
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.

Infinitesimal

infinitesimalsinfinitely closeinfinitesimally
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.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
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.

Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor.

Differential geometry of surfaces

surfaceshape operatorsmooth surface
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.

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
Explicit expressions for the Ricci tensor in terms of the Levi-Civita connection is given in the article on Christoffel symbols. 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) \nabla and Lie bracket by the following formula: 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.

Lie derivative

Lie bracketLie commutatorcommuting vector fields
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) \nabla and Lie bracket by the following formula:

Gregorio Ricci-Curbastro

RicciGregorio Ricci CurbastroRicci-Curbastro
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.

Connection form

connectionconnection one-formvertical subspace
The connection form gives an alternative way to describe curvature.

Vector bundle

vector bundlesWhitney sumdirect sum
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.

Principal bundle

principalprincipal ''G''-bundleprincipal fiber bundle
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.

Skew-symmetric matrix

skew-symmetricskew-symmetric matricesantisymmetric matrix
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, the Lie algebra of the orthogonal group, which is the structure group of the tangent bundle of a Riemannian manifold).

Differential form

differential forms2-formtwo-form
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, the Lie algebra of the orthogonal group, which is the structure group of the tangent bundle of a Riemannian manifold).

Lie algebra

Lie bracketLie algebrasabelian Lie algebra
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, the Lie algebra of the orthogonal group, which is the structure group of the tangent bundle of a Riemannian manifold).

Orthogonal group

special orthogonal grouprotation grouporthogonal
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, the Lie algebra of the orthogonal group, which is the structure group of the tangent bundle of a Riemannian manifold).