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The Poincaré Lemma in differential geometry and complex analytic geometry asserts that “every differential form $\omega$ which is closed, $d_{dR}\omega = 0$, is locally exact, $\omega|_U = d_{dR}\kappa$.”
In more detail: if $X$ is contractible then for every closed differential form $\omega \in \Omega^k_{cl}(X)$ with $k \geq 1$ there exists a differential form $\lambda \in \Omega^{k-1}(X)$ such that
Moreover, for $\omega$ a smooth family of closed forms, there is a smooth family of $\lambda$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
$f_0, f_1 \,\colon\, X \to Y$ be a pair of smooth functions between smooth manifolds
$\Psi \colon [0,1] \times X \to Y$ 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 $X$ and $Y$, an explicit formula for which is given by the following “homotopy operator”:
In particular, $f_0^*$ and $f_1^*$ coincide on de Rham cohomology:
Here $\iota_{\partial t}$ denotes contraction (cf. Cartan calculus) with the canonical vector field tangent to $[0,1]$, 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 $f_2 = const_y$ is a function constant on a point $y \in Y$:
If a smooth manifold $X$ admits a smooth contraction
then the de Rham cohomology of $X$ is concentrated on the ground field in degree 0. Moreover, for $\omega$ any closed form on $X$ in positive degree, an explicit formula for a form $\lambda$ with $d \lambda = \omega$ is given by
More generally, the conclusion of the Poincaré lemma for differential forms of bounded degree $\leq n$ follows already on $n$-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 $\omega$ is exact, with potential function given at $x \in X$ by the integral of $\omega$ from any fixed base point along any smooth path to $x$.
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 account:
Course notes:
Discussion in complex analytic geometry:
following
Last revised on January 3, 2024 at 14:22:26. See the history of this page for a list of all contributions to it.