Poincaré lemma

The Poincaré lemma states that every closed differential form is locally exact (http://planetmath.org/ExactDifferentialForm).


(Poincaré Lemma) [1] Suppose X is a smooth manifoldMathworldPlanetmath, Ωk(X) is the set of smooth differential k-forms on X, and suppose ω is a closed form in Ωk(X) for some k>0.

  • Then for every xX there is a neighbourhood UX, and a (k-1)-form ηΩk-1(U), such that


    where ι is the inclusion ι:UX.

  • If X is contractible, this η exists globally; there exists a (k-1)-form ηΩk-1(X) such that



Despite the name, the Poincaré lemma is an extremely important result. For instance, in algebraic topology, the definition of the kth de Rham cohomology group


can be seen as a measure of the degree in which the Poincaré lemma fails. If Hk(X)=0, then every k form is exact, but if Hk(X) is non-zero, then X has a non-trivial topology (or “holes”) such that k-forms are not globally exact. For instance, in X=2{0} with polar coordinatesMathworldPlanetmath (r,ϕ), the 1-form ω=dϕ is not globally exact.


  • 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
Title Poincaré lemma
Canonical name PoincareLemma
Date of creation 2013-03-22 14:06:28
Last modified on 2013-03-22 14:06:28
Owner matte (1858)
Last modified by matte (1858)
Numerical id 12
Author matte (1858)
Entry type Theorem
Classification msc 53-00
Related topic ExactDifferentialForm
Related topic ClosedDifferentialFormsOnASimpleConnectedDomain
Related topic LaminarField