Poincaré lemma

The Poincaré lemma states that every closed differential form of positive degree is locally exact. On a star-shaped domain, every closed form of positive degree is exact.

Exterior derivative: The differential operator on differential forms that generalizes the gradient, curl, and divergence. Closedness and exactness are defined by whether this operator gives zero or produces the form.
Differential form: A smooth field of alternating multilinear functions on tangent vectors, integrated over oriented subspaces. The lemma applies to forms of positive degree.
de Rham cohomology: The cohomology theory formed from closed differential forms modulo exact forms. The lemma makes its positive-degree classes vanish on sufficiently small neighborhoods.
Henri Poincaré: A French mathematician and physicist whose work shaped topology, celestial mechanics, and mathematical physics. The lemma bears his name and reflects his foundational work in topology.
Closed form: A differential form whose exterior derivative is zero. The lemma’s hypothesis is precisely that the form is closed.
Differential k-form: A differential form that takes k tangent vectors as arguments. The lemma applies degree by degree to forms with k greater than zero.
Poincaré lemma with parameters: A parameter-dependent version of the Poincaré lemma that produces primitives varying smoothly with parameters. It extends the local primitive construction to smoothly varying families of forms.
Élie Cartan: A French mathematician who developed influential methods in differential geometry and Lie theory. Cartan’s calculus of differential forms helped establish the lemma’s modern geometric setting.
Exact form: A differential form that is the exterior derivative of another form. The lemma’s conclusion is that a closed form has a primitive locally.
Manifold: A space that locally resembles Euclidean space and supports consistent smooth coordinates. Local exactness is asserted on smooth manifolds.

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