Editing Connection (vector bundle) (section)

=== 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.