Editing Connection (vector bundle) (section)

==Curvature==

{{See also|Curvature form}}

The '''curvature''' of a connection <math>\nabla</math> on <math>E\to M</math> is a 2-form <math>F_{\nabla}</math> on <math>M</math> with values in the endomorphism bundle <math>\operatorname{End}(E) = E^* \otimes E</math>. That is,
:<math>F_\nabla \in \Omega^2(\mathrm{End}(E)) = \Gamma(\Lambda^2T^*M \otimes \mathrm{End}(E)).</math>
It is defined by the expression
:<math>F_\nabla(X,Y)(s) = \nabla_X\nabla_Y s- \nabla_Y\nabla_X s- \nabla_{[X,Y]}s</math>
where <math>X</math> and <math>Y</math> are tangent vector fields on <math>M</math> and <math>s</math> is a section of <math>E</math>. One must check that <math>F_{\nabla}</math> is [[Smooth function|<math>C^{\infty}(M)</math>]]-linear in both <math>X</math> and <math>Y</math> and that it does in fact define a bundle endomorphism of <math>E</math>.

As mentioned [[#Exterior_covariant_derivative_and_vector-valued_forms|above]], the covariant exterior derivative <math>d_{\nabla}</math> need not square to zero when acting on <math>E</math>-valued forms. The operator <math>d_{\nabla}^2</math> is, however, strictly tensorial (i.e. <math>C^{\infty}(M)</math>-linear). This implies that it is induced from a 2-form with values in <math>\operatorname{End}(E)</math>. This 2-form is precisely the curvature form given above. For an <math>E</math>-valued form <math>\sigma</math> we have
:<math>(d_\nabla)^2\sigma = F_\nabla\wedge\sigma.</math>

A '''flat connection''' is one whose curvature form vanishes identically.

=== Local form and Cartan's structure equation ===

The curvature form has a local description called '''Cartan's structure equation'''. If <math>\nabla</math> has local form <math>A</math> on some trivialising open subset <math>U\subset M</math> for <math>E</math>, then
:<math>F_{\nabla} = dA + A \wedge A</math>
on <math>U</math>. To clarify this notation, notice that <math>A</math> is a endomorphism-valued one-form, and so in local coordinates takes the form of a matrix of one-forms. The operation <math>d</math> applies the exterior derivative component-wise to this matrix, and <math>A\wedge A</math> denotes matrix multiplication, where the components are wedged rather than multiplied.

In local coordinates <math>\mathbf{x} = (x^1,\dots,x^n)</math> on <math>M</math> over <math>U</math>, if the connection form is written <math>A=A_\ell dx^\ell = (\Gamma_{\ell i}^{\ \ j}) dx^\ell</math> for a collection of local endomorphisms <math>A_\ell = (\Gamma_{\ell i}^{\ \ j})</math>, then one has

:<math>F_{\nabla} = \sum_{p,q=1}^n \frac{1}{2} \left( \frac{\partial A_q}{\partial x^p} - \frac{\partial A_p}{\partial x^q} + [A_p, A_q]\right) dx^p \wedge dx^q.</math>

Further expanding this in terms of the Christoffel symbols <math>\Gamma_{\ell i}^{\ \ j}</math> produces the familiar expression from Riemannian geometry. Namely if <math>s=s^i e_i</math> is a section of <math>E</math> over <math>U</math>, then

:<math>F_{\nabla}(s) = \sum_{i,j=1}^k \sum_{p,q=1}^n \frac{1}{2} \left( \frac{\partial \Gamma_{qi}^{\ \ j}}{\partial x^p} - \frac{\partial \Gamma_{pi}^{\ \ j}}{\partial x^q} + \Gamma_{pr}^{\ \ j} \Gamma_{qi}^{\ \ r} - \Gamma_{qr}^{\ \ j} \Gamma_{pi}^{\ \ r} \right) s^i dx^p \wedge dx^q \otimes e_j = \sum_{i,j=1}^k \sum_{p,q=1}^n R_{pqi}^{\ \ \ j} s^i dx^p\wedge dx^q \otimes e_j.</math>
Here <math>R=(R_{pqi}^{\ \ \ j})</math> is the full '''curvature tensor''' of <math>F_{\nabla}</math>, and in Riemannian geometry would be identified with the [[Riemannian curvature tensor]].

It can be checked that if we define <math>[A, A]</math> to be wedge product of forms but [[commutator]] of endomorphisms as opposed to composition, then <math>A \wedge A = \frac{1}{2} [A, A]</math>, and with this alternate notation the Cartan structure equation takes the form
:<math>F_{\nabla} = dA + \frac{1}{2} [A, A].</math>
This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection form <math>\omega</math>, a [[Lie algebra]]-valued one-form, for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.

In some references (see for example {{harv|MadsenTornehave1997}}) the Cartan structure equation may be written with a minus sign:
:<math>F_{\nabla} = dA - A \wedge A.</math>
This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.

===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.