coboundary definition of exterior derivative
Let be a smooth manifold, and
-
•
let denote the algebra
of smooth functions
on ;
-
•
let denote the Lie-algebra of smooth vector fields;
-
•
and let denote the vector space
of smooth, differential
-forms.
Recall that a differential form is a
multilinear, alternating
mapping
such that, in local coordinates, looks like a multilinear combination of its vector field arguments. Thus, employing the Einstein summation convention and local coordinates , we have
where is a list of vector fields. Recall also
that is a module. The action is given by
a directional derivative, and takes the form
With these preliminaries out of the way, we have the following description of the exterior derivative operator . For , we have
(1) | ||||
where indicates the omission of the argument .
The above expression (1) of can be taken as
the definition of the exterior derivative. Letting the arguments be coordinate vector fields, it is not hard to show that the above definition is equivalent to the
usual definition of as a derivation of the exterior algebra of
differential forms, or the local coordinate definition of . 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
, is just
zeroth-order Lie algebra cohomology of with coefficients in
. 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 |