# coboundary definition of exterior derivative

Let $M$ be a smooth manifold  , and

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

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

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

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

 $v(f)=v^{i}\partial_{i}f,\quad v\in V(M),\;f\in C^{\infty}(M).$

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

 $\displaystyle(d\omega)(v_{0},v_{1},\dots,v_{k})=$ $\displaystyle\sum_{0\leq i\leq k}(-1)^{k}v_{i}\omega(\dots,\widehat{v}_{i},% \dots)+$ (1) $\displaystyle\ +\sum_{0\leq i

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   $d:\Omega^{k}(M)\to\Omega^{k+1}(M)$, is just zeroth-order Lie algebra cohomology of $V(M)$ with coefficients in $C^{\infty}(M)$. 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 CoboundaryDefinitionOfExteriorDerivative 2013-03-22 15:38:06 2013-03-22 15:38:06 rmilson (146) rmilson (146) 15 rmilson (146) Definition msc 15A69 msc 58A10 LieAlgebraCohomology