In differential geometry, the **curvature form** describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.

Let *G* be a Lie group with Lie algebra , and *P* → *B* be a principal *G*-bundle. Let ω be an Ehresmann connection on *P* (which is a -valued one-form on *P*).

Then the **curvature form** is the -valued 2-form on *P* defined by

(In another convention, 1/2 does not appear.) Here stands for exterior derivative, is defined in the article "Lie algebra-valued form" and *D* denotes the exterior covariant derivative. In other terms,^{[1]}

where *X*, *Y* are tangent vectors to *P*.

There is also another expression for Ω: if *X*, *Y* are horizontal vector fields on *P*, then^{[2]}

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.

A connection is said to be **flat** if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology.

If *E* → *B* is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan:

where is the wedge product. More precisely, if and denote components of ω and Ω correspondingly, (so each is a usual 1-form and each is a usual 2-form) then

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. In this case the form Ω is an alternative description of the curvature tensor, i.e.

using the standard notation for the Riemannian curvature tensor.

If is the canonical vector-valued 1-form on the frame bundle, the torsion of the connection form is the vector-valued 2-form defined by the structure equation

where as above *D* denotes the exterior covariant derivative.

The first Bianchi identity takes the form

The second Bianchi identity takes the form

and is valid more generally for any connection in a principal bundle.

The Bianchi identities can be written in tensor notation as:

The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of general theory of relativity.^{[clarification needed]}

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

- Connection (principal bundle)
- Basic introduction to the mathematics of curved spacetime
- Contracted Bianchi identities
- Einstein tensor
- Einstein field equations
- General theory of relativity
- Chern-Simons form
- Curvature of Riemannian manifolds
- Gauge theory

Various notions of curvature defined in differential geometry | |
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Differential geometry of curves | |

Differential geometry of surfaces | |

Riemannian geometry | |

Curvature of connections |