coboundary definition of exterior derivative

Let M be a smooth manifoldMathworldPlanetmath, and

Recall that a differential formMathworldPlanetmath αΩk(M) is a multilinear, alternatingPlanetmathPlanetmath mapping

α:V(M)××V(M)(k times)C(M)

such that, in local coordinates, α looks like a multilinear combination of its vector field arguments. Thus, employing the Einstein summation convention and local coordinates , we have


where u,v,,w is a list of k vector fields. Recall also that C(M) is a V(M) module. The action is given by a directional derivativeMathworldPlanetmathPlanetmath, and takes the form


With these preliminaries out of the way, we have the following description of the exterior derivative operator d:Ωk(M)Ωk+1(M). For ωΩk(M), we have

(dω)(v0,v1,,vk)= 0ik(-1)kviω(,v^i,)+ (1)

where v^i indicates the omission of the argument vi.

The above expression (1) of dω can be taken as the definition of the exterior derivative. Letting the vi arguments be coordinate vector fields, it is not hard to show that the above definition is equivalent to the usual definition of d as a derivation of the exterior algebra of differential forms, or the local coordinate definition of d. The nice feature of (1) is that it is equivalent to the definition of the coboundary operator for Lie algebra cohomology. Thus, we see that de Rham cohomologyMathworldPlanetmath, which is the cohomology of the cochain complexMathworldPlanetmathPlanetmath d:Ωk(M)Ωk+1(M), is just zeroth-order Lie algebra cohomology of V(M) with coefficients in C(M). The bit about “zeroth order” means that we are considering cochains that are zeroth order differential operatorsMathworldPlanetmath of their arguments — in other words, differential forms.

Title coboundary definition of exterior derivative
Canonical name CoboundaryDefinitionOfExteriorDerivative
Date of creation 2013-03-22 15:38:06
Last modified on 2013-03-22 15:38:06
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 15
Author rmilson (146)
Entry type Definition
Classification msc 15A69
Classification msc 58A10
Related topic LieAlgebraCohomology