# Curvature form

Bianchi identityBianchi identitiescurvatureflat connectioncurvature 2-formBianchi's second identityflatcurvature tensorcurvature two-formsecond Bianchi identity
In differential geometry, the curvature form describes curvature of a connection on a principal bundle.wikipedia
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### Connection form

connectionconnection one-formvertical subspace
In differential geometry, the curvature form describes curvature of a connection on a principal bundle.
The main tensorial invariant of a connection form is its curvature form.

### Curvature

curvednegative curvatureextrinsic curvature
In differential geometry, the curvature form describes curvature of a connection on a principal bundle.
Various generalizations capture in an abstract form this idea of curvature as a measure of holonomy; see curvature form.

### Ehresmann connection

connectionEhresmannhorizontal lift
Let ω be an Ehresmann connection on P (which is a \mathfrak g-valued one-form on P).
Furthermore, the connection form allows for a definition of curvature as a curvature form as well.

### Flat vector bundle

flatflatness
In mathematics, a vector bundle is said to be flat if it is endowed with a linear connection with vanishing curvature, i.e. a flat connection.

### Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
It can be considered as an alternative to or generalization of the curvature tensor in Riemannian geometry.
On a Riemannian manifold one has the covariant derivative \nabla_u R and the Bianchi identity (often called the second Bianchi identity or differential Bianchi identity) takes the form:

### Exterior covariant derivative

covariant exterior derivativeexterior covariant differentiation
Here d stands for exterior derivative, is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative.
where F = ρ is the representation in of the curvature two-form Ω.

### Differential form

differential forms2-formtwo-form
Let ω be an Ehresmann connection on P (which is a \mathfrak g-valued one-form on P).
This form is a special case of the curvature form on the

### Connection (principal bundle)

principal connectionconnectionconnections
The curvature form of a principal G-connection ω is the \mathfrak g-valued 2-form Ω defined by

### Gauge theory

gauge groupgauge theoriesgauge symmetry
The curvature form F, a Lie algebra-valued 2-form that is an intrinsic quantity, is constructed from a connection form by

### Curvature of Riemannian manifolds

curvatureabstract definition of curvaturecurvature of a Riemannian manifold

### Differential geometry

differentialdifferential geometerdifferential geometry and topology
In differential geometry, the curvature form describes curvature of a connection on a principal bundle.

### Principal bundle

principalprincipal ''G''-bundleprincipal fiber bundle
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. Let G be a Lie group with Lie algebra \mathfrak g, and P → B be a principal G-bundle.

### Riemannian geometry

Riemannianlocal to global theoremsRiemann geometry
It can be considered as an alternative to or generalization of the curvature tensor in Riemannian geometry.

### Lie group

Lie groupsLie subgroupmatrix Lie group
Let G be a Lie group with Lie algebra \mathfrak g, and P → B be a principal G-bundle.

### Lie algebra

Lie bracketLie algebrasabelian Lie algebra
Let G be a Lie group with Lie algebra \mathfrak g, and P → B be a principal G-bundle.

### Lie algebra-valued differential form

Lie algebra-valued formLie algebra-valued forms-valued form
Let ω be an Ehresmann connection on P (which is a \mathfrak g-valued one-form on P). Here d stands for exterior derivative, is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative.

### Exterior derivative

exterior calculusexterior differentiationdifferentials
Here d stands for exterior derivative, is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative.

### Fundamental vector field

motivationthe vector field on ''P'' associated to ''ξ'' by differentiating the ''G'' action on ''P
:where hZ means the horizontal component of Z, on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and is the inverse of the normalization factor used by convention in the formula for the exterior derivative.

### Exterior algebra

exterior productexterior powerwedge product
where \wedge is the wedge product.

### Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices.

### Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices.

### Skew-symmetric matrix

skew-symmetricskew-symmetric matricesantisymmetric matrix
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices.

### Shoshichi Kobayashi

KobayashiKobayashi, Shoshichi
* Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.

### Katsumi Nomizu

NomizuNomizu, Katsumi
* Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.

### Foundations of Differential Geometry

* Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.