# 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:

### Curvature form

**Bianchi identityBianchi identitiescurvature**

### 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**

### 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**