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The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form which is closed, , is locally exact, .”
In more detail: if is contractible then for every closed differential form with there exists a differential form such that
Moreover, for a smooth family of closed forms, there is a smooth family of s satisfying this condition.
This statement has several more abstract incarnations. One is that it says that on a Cartesian space (or a complex polydisc) the de Rham cohomology (the holomorphic de Rham cohomology) vanishes in positive degree.
Still more abstractly this says that the canonical morphisms of sheaves of chain complexes
from the locally constant sheaf on the real numbers (the complex numbers) to the de Rham complex (holomorphic de Rham complex) is a stalk-wise quasi-isomorphism – hence an equivalence in the derived category and hence induce an equivalence in hyper-abelian sheaf cohomology. (The latter statement fails in general in complex algebraic geometry, see (Illusie 12, 1.) and see also at GAGA.) (A variant of such resolutions of constant sheaves for the case over Klein geometries are BGG resolutions.)
The Poincaré lemma is a special case of the more general statement that the pullbacks of differential forms along homotopic smooth function are related by a chain homotopy.
Let
be a pair of smooth functions between smooth manifolds
a smooth homotopy between them.
Then there is a chain homotopy between the induced operations of pullback of differential forms:
on the de Rham complexes of and , an explicit formula for which is given by the following “homotopy operator”:
In particular, and coincide on de Rham cohomology:
Here denotes contraction (cf. Cartan calculus) with the canonical vector field tangent to , and the integration is that of functions with values in the vector space of differential forms.
We compute as follows:
where in the second step we used that exterior differential commutes with pullback of differential forms (this Prop.), and in the last step the Stokes theorem.
The Poincaré lemma proper is the special case of this statement for the case that is a function constant on a point :
If a smooth manifold admits a smooth contraction
then the de Rham cohomology of is concentrated on the ground field in degree 0. Moreover, for any closed form on in positive degree, an explicit formula for a form with is given by
More generally, the conclusion of the Poincaré lemma for differential forms of bounded degree follows already on -connected spaces (for instance by combining the Hurewicz theorem first with the universal coefficient theorem and then with the de Rham theorem).
Explicitly:
On a simply-connected (i.e.: 1-connected) smooth manifold, a closed differential 1-form is exact, with potential function given at by the integral of from any fixed base point along any smooth path to .
This follows locally for instance by the fiberwise Stokes theorem (here) and then globally due to the independence of the choice of path, by the assumption of simple-connectivity and the plain Stokes theorem.
Textbook accounts which make this explicit include do Carmo 1994, Prop. 3 (p. 24) in §3. Exposition is also in Armstrong 2017.
Textbook accounts:
Dominic G. B. Edelen, Lem 4-1.2 and §5-3 in: Applied exterior calculus, Wiley (1985) [GoogleBooks]
Manfredo P. do Carmo, §4.3 of Differential Forms and Applications, Springer (1994) [doi:10.1007/978-3-642-57951-6]
Course notes:
Litt: An expository account of differential forms and the Poincaré Lemma using modern methods, aimed at beginning undergraduates. Contains some minor errors and omissions (in the exterior power section).
Discussion in complex analytic geometry:
following
Generalization to supermanifolds
for plain differential forms (where the Lemma and its proof remain formally the same):
and for (compactly supported) integral forms:
Generalization to non-abelian Lie 2-algebra valued differential forms (local connections on 2-bundles):
Generalization to covariant derivatives:
Last revised on August 26, 2024 at 17:08:57. See the history of this page for a list of all contributions to it.