Cartan Homotopy Formula

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 $\iota_X \omega$ is sometimes written as $X \mathbin \omega.$

Definition

The interior product is defined to be the contraction of a differential form with a vector field. Thus if $X$ is a vector field on the manifold $M,$ then
$\iota_X : \Omega^p(M) \to \Omega^(M)$
is the map which sends a $p$-form $\omega$ to the $(p - 1)$-form $\iota_X \omega$ defined by the property that
$(\iota_X\omega)\left(X_1, \ldots, X_\right) = \omega\left(X, X_1, \ldots, X_\right)$
for any vector fields $X_1, \ldots, X_.$

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms $\alpha$
$\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha, X \rangle,$
where $\langle \,\cdot, \cdot\, \rangle$ is the duality pairing between $\alpha$ and the vector $X.$ Explicitly, if $\beta$ is a $p$-form and $\gamma$ is a $q$-form, then
$\iota_X(\beta \wedge \gamma) = \left(\iota_X\beta\right) \wedge \gamma + (-1)^p \beta \wedge \left(\iota_X\gamma\right).$
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

If in local coordinates $(x_1,...,x_n)$ the vector field $X$ is described by functions $f_1,...,f_n$, then the interior product is given by
$\iota_X (dx_1 \wedge ...\wedge dx_n) = \sum_^(-1)^f_r dx_1 \wedge ...\wedge \widehat \wedge ... \wedge dx_n,$
where $dx_1\wedge ...\wedge \widehat \wedge ... \wedge dx_n$ is the form obtained by omitting $dx_r$ from $dx_1 \wedge ...\wedge dx_n$.

By antisymmetry of forms,
$\iota_X \iota_Y \omega = - \iota_Y \iota_X \omega,$
and so $\iota_X \circ \iota_X = 0.$ This may be compared to the exterior derivative $d,$ which has the property $d \circ d = 0.$

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 or Cartan magic formula):
$\mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\ \omega.$

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. The Cartan homotopy formula is named after Élie Cartan.

The interior product with respect to the commutator of two vector fields $X,$ $Y$ satisfies the identity
\iota_ = \left mathcal_X, \iota_Y\right

See also

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Notes

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.

{{DEFAULTSORT:Interior Product
Differential forms
Differential geometry
Multilinear algebra