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Interior product

## Summary

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product. The interior product ${\displaystyle \iota _{X}\omega }$ is sometimes written as ${\displaystyle X\mathbin {\lrcorner } \omega .}$[1]

## Definition

The interior product is defined to be the contraction of a differential form with a vector field. Thus if ${\displaystyle X}$  is a vector field on the manifold ${\displaystyle M,}$  then

${\displaystyle \iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)}$

is the map which sends a ${\displaystyle p}$ -form ${\displaystyle \omega }$  to the ${\displaystyle (p-1)}$ -form ${\displaystyle \iota _{X}\omega }$  defined by the property that
${\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)}$

for any vector fields ${\displaystyle X_{1},\ldots ,X_{p-1}.}$

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms ${\displaystyle \alpha }$

${\displaystyle \displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,}$

where ${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$  is the duality pairing between ${\displaystyle \alpha }$  and the vector ${\displaystyle X.}$  Explicitly, if ${\displaystyle \beta }$  is a ${\displaystyle p}$ -form and ${\displaystyle \gamma }$  is a ${\displaystyle q}$ -form, then
${\displaystyle \iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).}$

The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.

## Properties

By antisymmetry of forms,

${\displaystyle \iota _{X}\iota _{Y}\omega =-\iota _{Y}\iota _{X}\omega }$

and so ${\displaystyle \iota _{X}\circ \iota _{X}=0.}$  This may be compared to the exterior derivative ${\displaystyle d,}$  which has the property ${\displaystyle d\circ d=0.}$

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (also known as the Cartan identity, Cartan homotopy formula[2] or Cartan magic formula):

${\displaystyle {\mathcal {L}}_{X}\omega =d(\iota _{X}\omega )+\iota _{X}d\omega =\left\{d,\iota _{X}\right\}\omega .}$

This identity defines a duality between the exterior and interior derivatives. Cartan's identity is important in symplectic geometry and general relativity: see moment map.[3] The Cartan homotopy formula is named after Élie Cartan.[4]

The interior product with respect to the commutator of two vector fields ${\displaystyle X,}$  ${\displaystyle Y}$  satisfies the identity

${\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right].}$

## Notes

1. ^ The character ⨼ is U+2A3C in Unicode
2. ^ Tu, Sec 20.5.
3. ^ There is another formula called "Cartan formula". See Steenrod algebra.
4. ^ Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25

## References

• Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
• Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011. doi:10.1007/978-1-4419-7400-6