Curvature form

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.

Definition

Let G be a Lie group with Lie algebra <math>\mathfrak g</math>, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a <math>\mathfrak g</math>-valued one-form on P).

Then the curvature form is the <math>\mathfrak g</math>-valued 2-form on P defined by

:<math>\Omega=d\omega + {1 \over 2}[\omega \wedge \omega] = D \omega.</math>

(In another convention, 1/2 does not appear.) Here <math>d</math> stands for exterior derivative, <math>[\cdot \wedge \cdot]</math> is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

:<math>\,\Omega(X, Y)= d\omega(X,Y) + {1 \over 2}[\omega(X),\omega(Y)]</math>

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

:<math>\sigma\Omega(X, Y) = -\omega([X, Y]) = -[X, Y] + h[X, Y]</math>

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 <math>\sigma\in \{1, 2\}</math> 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.

Curvature form in a vector bundle

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:

:<math>\,\Omega = d\omega + \omega \wedge \omega, </math>

where <math>\wedge</math> is the wedge product. More precisely, if <math>{\omega^i}_j</math> and <math>{\Omega^i}_j</math> denote components of ω and Ω correspondingly, (so each <math>{\omega^i}_j</math> is a usual 1-form and each <math>{\Omega^i}_j</math> is a usual 2-form) then

:<math>\Omega^i_j = d{\omega^i}_j + \sum_k {\omega^i}_k \wedge {\omega^k}_j.</math>

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.

:<math>\,R(X, Y) = \Omega(X, Y),</math>

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

If <math>\theta</math> is the canonical vector-valued 1-form on the frame bundle, the torsion <math>\Theta</math> of the connection form <math>\omega</math> is the vector-valued 2-form defined by the structure equation

:<math>\Theta = d\theta + \omega\wedge\theta = D\theta,</math>

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

:<math>D\Theta = \Omega\wedge\theta.</math>

The second Bianchi identity takes the form

:<math>\, D \Omega = 0 </math>

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, a key component in the general theory of relativity.

Notes

References

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

Definition

Curvature form in a vector bundle

Bianchi identities

Notes

References

See also