nLab
Cartan's homotopy formula

Let M be a differentiable manifold, X a vector field on M, and X the Lie derivative along X. The contraction of a vector field and a k-form ω is denoted in modern literature by Xω (should be lrcorner instead of rfloor LaTeX command, but it does not work in iTeX) or ι Xω or ι(X)(ω).

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

Xω=dι(X)ω+ι(X)dω\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.

Regarding that the Cartan’s formula can be viewed as a formula about the de Rham complex, which has generalizations, one can often define the Cartan’s formulas for those generalizations. For example

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

Revised on November 29, 2012 05:05:43 by Zoran Škoda (193.55.36.18)