Curvature form

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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 g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}-valued one-form on P).

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

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

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

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

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]

σ Ω ( X , Y ) = − ω ( [ X , Y ] ) = − [ X , Y ] + h [ X , Y ] {\displaystyle \sigma \Omega (X,Y)=-\omega ([X,Y])=-[X,Y]+h[X,Y]} {\displaystyle \sigma \Omega (X,Y)=-\omega ([X,Y])=-[X,Y]+h[X,Y]}

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 σ ∈ { 1 , 2 } {\displaystyle \sigma \in \{1,2\}} {\displaystyle \sigma \in \{1,2\}} 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:

Ω = d ω + ω ∧ ω , {\displaystyle \,\Omega =d\omega +\omega \wedge \omega ,} {\displaystyle \,\Omega =d\omega +\omega \wedge \omega ,}

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

Ω j i = d ω i j + ∑ k ω i k ∧ ω k j . {\displaystyle \Omega _{j}^{i}=d{\omega ^{i}}_{j}+\sum _{k}{\omega ^{i}}_{k}\wedge {\omega ^{k}}_{j}.} {\displaystyle \Omega _{j}^{i}=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 ) = Ω ( X , Y ) , {\displaystyle \,R(X,Y)=\Omega (X,Y),} {\displaystyle \,R(X,Y)=\Omega (X,Y),}

using the standard notation for the Riemannian curvature tensor.

Bianchi identities

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

Θ = d θ + ω ∧ θ = D θ , {\displaystyle \Theta =d\theta +\omega \wedge \theta =D\theta ,} {\displaystyle \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 Θ = Ω ∧ θ . {\displaystyle D\Theta =\Omega \wedge \theta .} {\displaystyle D\Theta =\Omega \wedge \theta .}

The second Bianchi identity takes the form

D Ω = 0 {\displaystyle \,D\Omega =0} {\displaystyle \,D\Omega =0}

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

The Bianchi identities can be written in tensor notation as: R a b m n ; ℓ + R a b ℓ m ; n + R a b n ℓ ; m = 0. {\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.} {\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.}

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

  1. since [ ω ∧ ω ] ( X , Y ) = 1 2 ( [ ω ( X ) , ω ( Y ) ] − [ ω ( Y ) , ω ( X ) ] ) {\displaystyle [\omega \wedge \omega ](X,Y)={\frac {1}{2}}([\omega (X),\omega (Y)]-[\omega (Y),\omega (X)])} {\displaystyle [\omega \wedge \omega ](X,Y)={\frac {1}{2}}([\omega (X),\omega (Y)]-[\omega (Y),\omega (X)])}. Here we use also the σ = 2 {\displaystyle \sigma =2} {\displaystyle \sigma =2} Kobayashi convention for the exterior derivative of a one form which is then d ω ( X , Y ) = 1 2 ( X ω ( Y ) − Y ω ( X ) − ω ( [ X , Y ] ) ) {\displaystyle d\omega (X,Y)={\frac {1}{2}}(X\omega (Y)-Y\omega (X)-\omega ([X,Y]))} {\displaystyle d\omega (X,Y)={\frac {1}{2}}(X\omega (Y)-Y\omega (X)-\omega ([X,Y]))}
  2. Proof: σ Ω ( X , Y ) = σ d ω ( X , Y ) = X ω ( Y ) − Y ω ( X ) − ω ( [ X , Y ] ) = − ω ( [ X , Y ] ) . {\displaystyle \sigma \Omega (X,Y)=\sigma d\omega (X,Y)=X\omega (Y)-Y\omega (X)-\omega ([X,Y])=-\omega ([X,Y]).} {\displaystyle \sigma \Omega (X,Y)=\sigma d\omega (X,Y)=X\omega (Y)-Y\omega (X)-\omega ([X,Y])=-\omega ([X,Y]).}

References

See also