Cartan formula

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

Cartan formula in differential geometry

Further information: Interior product

The Cartan formula in differential geometry states:

${\displaystyle {\mathcal {L}}_{X}=\mathrm {d} \,\iota _{X}+\iota _{X}\mathrm {d} }$, where ${\displaystyle {\mathcal {L}}_{X},\mathrm {d} }$, and ${\displaystyle \iota _{X}}$ 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.

Cartan formula in algebraic topology

Further information: Steenrod algebra

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

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

The name derives from Henri Cartan, son of Élie.

Footnotes

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

See also

• List of things named after Élie Cartan