In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed).
Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n.[1] The lemma was introduced by Henri Poincaré in 1886.
[2][3] Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in
In particular, it implies that the de Rham complex yields a resolution of the constant sheaf
on M. The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma.
It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration.
The Poincaré lemma is also a special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
A standard proof of the Poincaré lemma uses the homotopy invariance formula (cf.
[8][9] The Poincaré lemma can be proved by means of integration along fibers.
[10][11] (This approach is a straightforward generalization of constructing a primitive function by means of integration in calculus.)
The same proof in fact shows the Poincaré lemma for any contractible open subset U of a manifold.
Cartan's magic formula for Lie derivatives can be used to give a short proof of the Poincaré lemma.
In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.
[13] If ω = p dx + q dy is a closed 1-form on (a, b) × (c, d), then py = qx.
Set so that gx = p. Then h = f − g must satisfy hx = 0 and hy = q − gy.
So and hence Similarly, if Ω = r dx ∧ dy then Ω = d(a dx + b dy) with bx − ay = r. Thus a solution is given by a = 0 and It is also possible to give an inductive proof of Poincaré's lemma which does not use homotopical arguments.
Due to linear independence of the coordinate differentials, this equation implies that
of an open subset U of a manifold M is defined as the quotient vector space Hence, the conclusion of the Poincaré lemma is precisely that if
The Poincaré lemma thus says the rest of the sequence is exact too (since a manifold is locally diffeomorphic to an open subset of
In the language of homological algebra, it means that the de Rham complex determines a resolution of the constant sheaf
Once one knows the de Rham theorem, the conclusion of the Poincaré lemma can then be obtained purely topologically.
Especially in calculus, the Poincaré lemma is stated for a simply connected open subset
By the de Rham theorem (which follows from the Poincaré lemma for open balls),
The pull-back along a proper map preserve compact supports; thus, the same proof as the usual one goes through.
for complex differential forms, which refine the exterior derivative by the formula
It states: let V be a submanifold of a manifold M and U a tubular neighborhood of V. If
(So the proof actually goes through if U is not a tubular neighborhood but if U deformation-retracts to V with homotopy relative to V.)
In characteristic zero, the following Poincaré lemma holds for polynomial differential forms.
is defined by the usual way; i.e., the linearity and This version of the lemma is seen by a calculus-like argument.
-Poincaré lemma; namely, the use of the fundamental theorem of calculus is replaced by Cauchy's integral formula.