The Poincaré lemma states that every closed differential form is locally exact (http://planetmath.org/ExactDifferentialForm).
Then for every there is a neighbourhood , and a -form , such that
where is the inclusion .
If is contractible, this exists globally; there exists a -form such that
can be seen as a measure of the degree in which the Poincaré lemma fails. If , then every form is exact, but if is non-zero, then has a non-trivial topology (or “holes”) such that -forms are not globally exact. For instance, in with polar coordinates , the -form is not globally exact.
- 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
|Date of creation||2013-03-22 14:06:28|
|Last modified on||2013-03-22 14:06:28|
|Last modified by||matte (1858)|