Knowlpedia

Bianchi identity

The covariant exterior derivative of the curvature form of a connection vanishes.

Let π:P→M\pi:P\to M be a , and let ω∈Ω1(P;g)\omega \in \Omega^1(P;\mathfrak{g}) be a connection 11-form. The curvature is the g\mathfrak{g}-valued 22-form

Ω  =  dω  +  12[ω∧ω],\Omega \;=\; d\omega \;+\; \frac{1}{2}[\omega \wedge \omega],

where [ω∧ω][\omega\wedge\omega] denotes the wedge product combined with the on g\mathfrak{g}.

For a g\mathfrak g-valued form η\eta, define

Dωη  =  dη  +  [ω∧η].D_\omega \eta \;=\; d\eta \;+\; [\omega \wedge \eta].

The Bianchi identity states that

DωΩ=0.D_\omega \Omega = 0.
Remarks
  • DωΩ=0D_\omega \Omega=0 is the bundle/form version of the "second Bianchi identity." In tensor notation for a covariant derivative ∇\nabla on the base, it corresponds to the cyclic identity
    (∇XR)(Y,Z)+(∇YR)(Z,X)+(∇ZR)(X,Y)=0,(\nabla_X R)(Y,Z) + (\nabla_Y R)(Z,X) + (\nabla_Z R)(X,Y) = 0,
    where RR is the curvature tensor.
  • In Chern–Weil theory, the Bianchi identity implies that characteristic forms built from Ω\Omega are closed. Combined with the , this explains how these forms descend from PP to well-defined de Rham cohomology classes on MM.