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Curvature form

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Title: Curvature form  
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Curvature form

In differential geometry, the curvature form describes curvature of a connection on a principal bundle. It can be considered as an alternative to or generalization of the curvature tensor in Riemannian geometry.

Contents

  • Definition 1
    • Curvature form in a vector bundle 1.1
  • Bianchi identities 2
  • Notes 3
  • References 4
  • See also 5

Definition

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

Then the curvature form is the \mathfrak g-valued 2-form on P defined by

\Omega=d\omega +{1\over 2}[\omega \wedge \omega]=D\omega.

Here d stands for exterior derivative, [\cdot \wedge \cdot] is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,

\,\Omega(X,Y)=d\omega(X,Y) + {1 \over 2}([\omega(X),\omega(Y)] - [\omega(Y),\omega(X)])= d\omega(X,Y) + [\omega(X),\omega(Y)]

where X, Y are tangent vectors to P.

There is also another expression for Ω:

2\Omega(X, Y) = -[hX, hY] + h[X, Y]

where hZ means the horizontal component of Z and on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field).[1]

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. See also: flat vector bundle.

Curvature form in a vector bundle

If EB 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:

\,\Omega=d\omega +\omega\wedge \omega,

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

\Omega^i_{\ j}=d\omega^i_{\ j} +\sum_k \omega^i_{\ k}\wedge\omega^k_{\ j}.

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.

\,R(X,Y)=\Omega(X,Y),

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

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

\Theta=d\theta + \omega\wedge\theta = D\theta,

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

D\Theta=\Omega\wedge\theta.

The second Bianchi identity takes the form

\, D \Omega = 0

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

Notes

  1. ^ Proof: We can assume X, Y are either vertical or horizontal. Then we can assume X, Y are both horizontal (otherwise both sides vanish since Ω is horizontal). In that case, this is a consequence of the invariant formula for exterior derivative d and the fact ω(Z) is a unique Lie algebra element that generates the vector field Z.

References

See also

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