Cartan's homotopy formula

Let $M$ be a differentiable manifold, $X$ a vector field on $M$, and ${\mathcal{L}}_{X}$ the Lie derivative along $X$. The contraction of a vector field and a $k$-form $\omega $ is denoted in modern literature by $X\rfloor \omega $ (should be lrcorner instead of rfloor LaTeX command, but it does not work in iTeX) or ${\iota}_{X}\omega $ or $\iota (X)(\omega )$.

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.

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)