coboundary definition of exterior derivative

Let $M$ be a smooth manifold, and

• let $C^{\mathrm{\infty }}(M)$ denote the algebra of smooth functions on $M$;

• let $V(M)$ denote the Lie-algebra of smooth vector fields;

• and let ${\mathrm{\Omega }}^k(M)$ denote the vector space of smooth, differential $k$-forms.

Recall that a differential form $\alpha \in {\mathrm{\Omega }}^k(M)$ is a multilinear, alternating mapping

$$\alpha :V(M)\times \mathrm{\cdots }\times V(M)\text{(k times)}\to C^{\mathrm{\infty }}(M)$$

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

$$\alpha (u,v,\mathrm{\dots },w)={\alpha }_{ij\mathrm{\dots }k}{u}^i{v}^j\mathrm{\cdots }{w}^k,$$

where $u,v,\mathrm{\dots },w$ is a list of $k$ vector fields. Recall also that $C^{\mathrm{\infty }}(M)$ is a $V(M)$ module. The action is given by a directional derivative, and takes the form

$$v(f)=v^i{\partial }_{i}f,v\in V(M),f\in C^{\mathrm{\infty }}(M).$$

With these preliminaries out of the way, we have the following description of the exterior derivative operator $d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$. For $\omega \in {\mathrm{\Omega }}^k(M)$, we have

$(d\omega )(v_0,v_1,\mathrm{\dots },v_k)=$ ${\displaystyle \sum _{0\le i\le k}}{(-1)}^k{v}_i\omega (\mathrm{\dots },{\widehat{v}}_i,\mathrm{\dots })+$ (1)

where ${\widehat{v}}_i$ indicates the omission of the argument $v_i$.

The above expression (1) of $d\omega$ can be taken as the definition of the exterior derivative. Letting the $v_i$ 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 cohomology, which is the cohomology of the cochain complex $d:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{k+1}(M)$, is just zeroth-order Lie algebra cohomology of $V(M)$ with coefficients in $C^{\mathrm{\infty }}(M)$. The bit about “zeroth order” means that we are considering cochains that are zeroth order differential operators 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