Cartan formula

The Cartan formula in mathematics may refer to two different formulae in differential geometry or algebraic topology.

Cartan formula in differential geometry

The Cartan formula in differential geometry states:

{\mathcal {L}}_{X}=\mathrm {d} \,\iota _{X}+\iota _{X}\mathrm {d}

, where

{\mathcal {L}}_{X},\mathrm {d}

, and

\iota _{X}

It is also called the Cartan homotopy formula or Cartan magic formula. This formula is named after Élie Cartan.

The Cartan formula in algebraic topology is one of the five axioms of Steenrod algebra.^{[note 1]} It reads:

{\begin{aligned}Sq^{n}(x\smile y)&=\sum _{i+j=n}(Sq^{i}x)\smile (Sq^{j}y)\quad {\text{or}}\\P^{n}(x\smile y)&=\sum _{i+j=n}(P^{i}x)\smile (P^{j}y)\end{aligned}}

.

The name derives from Henri Cartan, son of Élie.

^ More precisely, these five axioms define cohomology operations, which are natural transformations between cohomology functors, which in turn define Steenrod algebra.