Cartan formula

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

The Cartan formula in differential geometry states:

${\displaystyle {\mathcal {L}}_{X}=\mathrm {d} ,円\iota _{X}+\iota _{X}\mathrm {d} }${\displaystyle {\mathcal {L}}_{X}=\mathrm {d} ,円\iota _{X}+\iota _{X}\mathrm {d} }, where ${\displaystyle {\mathcal {L}}_{X},\mathrm {d} }${\displaystyle {\mathcal {L}}_{X},\mathrm {d} }, and ${\displaystyle \iota _{X}}${\displaystyle \iota _{X}} are Lie derivative, exterior derivative, and interior product, respectively, acting on differential forms.

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:

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

• List of things named after Élie Cartan