proof of Poincaré lemma
Let be a smooth manifold, and let be a closed differential form of degree on . For any , there exists a contractible neighbourhood of (i.e. is homotopy equivalent to a single point), with inclusion map
To construct such a neighbourhood, take for example an open ball in a coordinate chart around . Because of the homotopy invariance of de Rham cohomology, the th de Rham cohomology group is isomorphic to that of a point; in particular,
Since , this implies that there exists a -form on such that . In the case where is a contractible manifold, such an exists globally since we can choose above.
|Title||proof of Poincaré lemma|
|Date of creation||2013-03-22 14:24:36|
|Last modified on||2013-03-22 14:24:36|
|Last modified by||pbruin (1001)|