Cartan's Magic Formula

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 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 $X_1, \ldots, X_.$

When $\omega$ is a scalar field (0-form), $\iota_X \omega = 0$ by convention.

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 given by

$X = f_1 \frac + \cdots + f_n \frac$

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 with respect to the commutator of two vector fields $X,$ $Y$ satisfies the identity
\iota_ = \left mathcal_X, \iota_Y\right= \left iota_X, \mathcal_Y\right Proof. For any k-form $\Omega$, $\mathcal L_X(\iota_Y \Omega) - \iota_Y (\mathcal L_X\Omega) = (\mathcal L_X\Omega)(Y, -) + \Omega(\mathcal L_X Y, -) - (\mathcal L_X \Omega)(Y , -) = \iota_\Omega = \iota_\Omega$and similarly for the other result.

Cartan identity

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):
$\mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\ \omega.$

where the anticommutator was used. 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.There is another formula called "Cartan formula". See Steenrod algebra. The Cartan homotopy formula is named after Élie Cartan.

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