synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Let be a differentiable manifold, a vector field on , and the Lie derivative along . Denote the contraction of a vector field and a differential form by .
Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says
The word “homotopy” is used because it supplies a homotopy operator for some manipulation with chain complexes in de Rham cohomology. Cartan’s homotopy formula is part of Cartan calculus.
See also noncommutative differential calculus where the formula is incorporated into the notion of Batalin-Vilkovisky module over a Gerstenhaber algebra.
The original reference:
See also:
Last revised on April 4, 2021 at 05:55:42. See the history of this page for a list of all contributions to it.