The Cartan formula in mathematics may refer to two different formulae in

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

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

are

Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...

exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...

, and

interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, contraction, or inner derivation) is a degree −1 (anti)derivation on the exterio ...

, respectively, acting on

differential forms In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...

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

Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. He ...

Cartan formula in algebraic topology

The Cartan formula in

is one of the five axioms of Steenrod algebra.More precisely, these five axioms define cohomology operations, which are

natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...

s between cohomology functors, which in turn define Steenrod algebra. It reads: :

\begin
Sq^n(x \smile y) & = \sum_ (Sq^i x) \smile (Sq^j y) \quad \text \\ P^n(x \smile y) & = \sum_ (P^i x) \smile (P^j y)
\end

. The name derives from

Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...

, son of Élie.

Footnotes

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Cartan formula in differential geometry

Cartan formula in algebraic topology

Footnotes

See also