Editing Connection (vector bundle) (section)

===Bianchi identity===

A version of the second (differential) [[Riemann_curvature_tensor#Symmetries_and_identities|Bianchi identity]] from Riemannian geometry holds for a connection on any vector bundle. Recall that a connection <math>\nabla</math> on a vector bundle <math>E\to M</math> induces an endomorphism connection on <math>\operatorname{End}(E)</math>. This endomorphism connection has itself an exterior covariant derivative, which we ambiguously call <math>d_{\nabla}</math>. Since the curvature is a globally defined <math>\operatorname{End}(E)</math>-valued two-form, we may apply the exterior covariant derivative to it. The '''Bianchi identity''' says that
:<math>d_{\nabla} F_{\nabla} = 0</math>.
This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.

There is no analogue in general of the ''first'' (algebraic) Bianchi identity for a general connection, as this exploits the special symmetries of the Levi-Civita connection. Namely, one exploits that the vector bundle indices of <math>E=TM</math> in the curvature tensor <math>R</math> may be swapped with the cotangent bundle indices coming from <math>T^*M</math> after using the metric to lower or raise indices. For example this allows the torsion-freeness condition <math>\Gamma_{\ell i}^{\ \ j} = \Gamma_{i \ell}^{\ \ j}</math> to be defined for the Levi-Civita connection, but for a general vector bundle the <math>\ell</math>-index refers to the local coordinate basis of <math>T^*M</math>, and the <math>i,j</math>-indices to the local coordinate frame of <math>E</math> and <math>E^*</math> coming from the splitting <math>\mathrm{End}(E)=E^* \otimes E</math>. However in special circumstance, for example when the rank of <math>E</math> equals the dimension of <math>M</math> and a [[solder form]] has been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.