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
See also: flat vector bundle.
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.