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 . The lemma was introduced by Henri Poincaré in 1886.
Informal discussion
Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in is exact.
In simpler terms, it means that if a differential form is closed in a region that can be shrunk to a point, then it can be written as the derivative of another form; i.e. if dα = 0 on a simply connected region, we can always find α = dβ; therefore we have d(dβ) = 0, expressed simply as d2 = 0. This concept is used in mathematical physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a 2-form) and you take its boundary (a 1-form, a curve), then the boundary of that boundary (a 0-form, a point) is an empty set.
In electromagnetism, magnetic fields can be described using a vector potential, and the Poincaré lemma helps in finding such potentials when the magnetic field is "well-behaved" (i.e., when the magnetic field is not due to a monopole), Gauss's law for magnetism states that the total magnetic flux through a closed surface is always zero, which implies that magnetic monopoles, if they exist, are not isolated but must be accompanied by other magnetic charges.
In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., ) vanishes for . 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.
Proofs A standard proof of the Poincaré lemma uses the homotopy invariance formula (cf. see the proofs below as well as Integration along fibers#Example)., and . The local form of the homotopy operator is described in and the connection of the lemma with the Maurer–Cartan form is explained in .
Direct proof The Poincaré lemma can be proved by means of integration along fibers. (This approach is a straightforward generalization of constructing a primitive function by means of integration in calculus.)
We shall prove the lemma for an open subset that is star-shaped or a cone over ; i.e., if is in , then is in for . This case in particular covers the open ball case, since an open ball can be assumed to be centered at the origin without loss of generality.
The trick is to consider differential forms on (we use for the coordinate on ). First define the operator (called the fiber integration) for k-forms on by
where , and similarly for and . Now, for , since , using the differentiation under the integral sign, we have:
where denote the restrictions of to the hyperplanes and they are zero since is zero there. If , then a similar computation gives . Thus, the above formula holds for any -form on . (The formula is a special case of a formula sometimes called the relative Stokes formula.)
Finally, let and then set . Then, with the notation , we get: for any -form on ,
the formula known as the homotopy formula. The operator is called the homotopy operator (also called a chain homotopy). Now, if is closed, . On the other hand, and , the latter because there is no nonzero higher form at a point. Hence,
which proves the Poincaré lemma.
The same proof in fact shows the Poincaré lemma for any contractible open subset U of a manifold. Indeed, given such a U, we have the homotopy with the identity and a point. Approximating such ,, we can assume is in fact smooth. The fiber integration is also defined for . Hence, the same argument goes through.
Proof using Lie derivatives Cartan's magic formula for Lie derivatives can be used to give a short proof of the Poincaré lemma. The formula states that the Lie derivative along a vector field is given as:
where denotes the interior product; i.e., .
Let be a smooth family of smooth maps for some open subset U of such that is defined for t in some closed interval I and is a diffeomorphism for t in the interior of I. Let denote the tangent vectors to the curve ; i.e., . For a fixed t in the interior of I, let . Then . Thus, by the definition of a Lie derivative, . That is,
Assume . Then, integrating both sides of the above and then using Cartan's formula and the differentiation under the integral sign, we get: for ,
where the integration means the integration of each coefficient in a differential form. Letting , we then have:
with the notation
Now, assume is an open ball with center ; then we can take . Then the above formula becomes: , which proves the Poincaré lemma when is closed.
Proof in the two-dimensional case In two dimensions the Poincaré lemma can be proved directly for closed 1-forms and 2-forms as follows.
If is a closed 1-form on , then . If then and . Set
so that . Then must satisfy and . The right hand side here is independent of x since its partial derivative with respect to x is . Since is independent of x, it is a function of y only. To find this function, we can evaluate it at any arbitrary value of x, specifically at the boundary :
and hence
Similarly, if then with . Thus a solution is given by and
Inductive proof It is also possible to give an inductive proof of Poincaré's lemma which does not use homotopical arguments. Let , where , be an m dimensional open coordinate cube. For each the projection which deletes the last m-n coordinates of a point is a fibration and therefore the pullback map is a monomorphism. We may thus consider any differential k-form which is defined on to be also defined on with no additional notation. Conversely, a k-form descends to a (necessarily unique) k-form on if and only if its expression with respect to the standard coordinates is
,
or in other words, its coefficients are functions of the variables only, and differentials of only the latter variables are involved.
For a differential k-form , let its codegree be the integer m-k. The induction is performed over the codegree of the form. Since we are working over a coordinate domain, partial derivatives and also integrals with respect to the coordinates can be applied to a form itself, by applying them to the coefficients of the form with respect to the canonical coordinates.
First let , i.e. the codegree is 0. It can be written as so if we define by , we havehence, is a primitive of .
Let now , where , i.e. has codegree m-k, and let us suppose that whenever a closed form has codegree less than m-k, the form is exact. The form can be decomposed aswhere neither nor contain any factor of . Define , then , where does not contain any factor of , hence, defining , this form is also closed, but does not involve any factor of . Since this form is closed, we havewhere the last term does not contain a factor of . Due to linear independence of the coordinate differentials, this equation implies thati.e. the form is a differential form in the variables only, hence it descends to an element of , and its codegree is thus m-k-1. The induction hypothesis applies, thus for some , thereforeconcluding the proof for a coordinate cube. In any manifold, every point has a neighborhood which is diffeomorphic to a coordinate cube, the proof also implies that on a manifold any closed k-form (for ) is locally exact.
Implication for de Rham cohomology By definition, the k-th de Rham cohomology group 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 is an open ball, then for . Now, differential forms determine a cochain complex called the de Rham complex:
where n = the dimension of M and denotes the sheaf of differential k-forms; i.e., consists of k-forms on U for each open subset U of M. It then gives rise to the complex (the augmented complex)
where is the constant sheaf with values in ; i.e., it is the sheaf of locally constant real-valued functions and the inclusion.
The kernel of is , since the smooth functions with zero derivatives are locally constant. Also, a sequence of sheaves is exact if and only if it is so locally. The Poincaré lemma thus says the rest of the sequence is exact too (since a manifold is locally diffeomorphic to an open subset of and then each point has an open ball as a neighborhood). In the language of homological algebra, it means that the de Rham complex determines a resolution of the constant sheaf . This then implies the de Rham theorem; i.e., the de Rham cohomology of a manifold coincides with the singular cohomology of it (in short, because the singular cohomology can be viewed as a sheaf cohomology.)
Once one knows the de Rham theorem, the conclusion of the Poincaré lemma can then be obtained purely topologically. For example, it implies a version of the Poincaré lemma for contractible or simply connected open sets (see §Simply connected case).
Simply connected case Especially in calculus, the Poincaré lemma is stated for a simply connected open subset . In that case, the lemma says that each closed 1-form on U is exact. This version can be seen using algebraic topology as follows. The rational Hurewicz theorem (or rather the real analog of that) says that since U is simply connected. Since is a field, the k-th cohomology is the dual vector space of the k-th homology . In particular, By the de Rham theorem (which follows from the Poincaré lemma for open balls), is the same as the first de Rham cohomology group (see §Implication to de Rham cohomology). Hence, each closed 1-form on U is exact.
Poincaré lemma with compact support There is a version of Poincaré lemma for compactly supported differential forms:
The usual proof in the non-compact case does not go through since the homotopy h is not proper. Thus, somehow a different argument is needed for the compact case.
Complex-geometry analog On complex manifolds, the use of the Dolbeault operators and for complex differential forms, which refine the exterior derivative by the formula , lead to the notion of -closed and -exact differential forms. The local exactness result for such closed forms is known as the Dolbeault–Grothendieck lemma (or -Poincaré lemma); cf. . Importantly, the geometry of the domain on which a -closed differential form is -exact is more restricted than for the Poincaré lemma, since the proof of the Dolbeault–Grothendieck lemma holds on a polydisk (a product of disks in the complex plane, on which the multidimensional Cauchy's integral formula may be applied) and there exist counterexamples to the lemma even on contractible domains.For counterexamples on contractible domains which have non-vanishing first Dolbeault cohomology see The -Poincaré lemma holds in more generality for pseudoconvex domains.
Using both the Poincaré lemma and the -Poincaré lemma, a refined local -Poincaré lemma can be proven, which is valid on domains upon which both the aforementioned lemmas are applicable. This lemma states that -closed complex differential forms are actually locally -exact (rather than just or -exact, as implied by the above lemmas).
Relative Poincaré lemma