nLab Cartan's homotopy formula

Statement

Let $M$ be a differentiable manifold, $X$ a vector field on $M$, and $\mathcal{L}_X$ the Lie derivative along $X$. Denote the contraction of a vector field and a differential form $\omega$ by $\iota(X)(\omega)$.

Then the Cartan’s infinitesimal homotopy formula, nowdays called simply Cartan’s homotopy formula or even Cartan formula, says

$$\mathcal{L}_X \omega = d \iota(X)\omega + \iota(X) d\omega$$

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.

Related concepts

• Cartan calculus

• Cartan's map

See also noncommutative differential calculus where the formula is incorporated into the notion of Batalin-Vilkovisky module over a Gerstenhaber algebra.

References

The original reference:

• Élie Cartan, Leçons sur les invariants intégraux (based on lectures given in 1920-21 in Paris, Hermann, Paris 1922, reprinted in 1958).

See also:

• Masoud Khalkhali, On Cartan homotopy formulas in cyclic homology, Manuscripta mathematica 94:1, pp 111-132 (1997) doi