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 <math>\iota_X \omega</math> is sometimes written as <math>\omega \mathbin{\lfloor} X</math>, which is called the right contraction of <math>\omega</math> with X.
Definition
The interior product is defined to be the contraction of a differential form with a vector field. Thus if <math>X</math> is a vector field on the manifold <math>M,</math> then
<math display=block>\iota_X : \Omega^p(M) \to \Omega^{p-1}(M)</math>
is the map which sends a <math>p</math>-form <math>\omega</math> to the <math>(p - 1)</math>-form <math>\iota_X \omega</math> defined by the property that
<math display=block>(\iota_X\omega)\left(X_1, \ldots, X_{p-1}\right) = \omega\left(X, X_1, \ldots, X_{p-1}\right)</math>
for any vector fields <math>X_1, \ldots, X_{p-1}.</math>
When <math>\omega</math> is a scalar field (0-form), <math>\iota_X \omega = 0</math> by convention.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms <math>\alpha</math>
<math display="block">\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha, X \rangle,</math>
where <math>\langle \,\cdot, \cdot\, \rangle</math> is the duality pairing between <math>\alpha</math> and the vector <math>X.</math> Explicitly, if <math>\alpha</math> is a <math>p</math>-form and <math>\beta</math> is a <math>q</math>-form, then
<math display="block">\iota_X(\alpha \wedge \beta) = \left(\iota_X\alpha\right) \wedge \beta + (-1)^p \alpha \wedge \left(\iota_X\beta\right).</math>
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 <math>(x_1, \ldots, x_n)</math> the vector field <math>X</math> is given by
<math>X = f_1 \frac{\partial}{\partial x_1} + \cdots + f_n \frac{\partial}{\partial x_n} </math>
then the interior product is given by
<math display="block">\iota_X (dx_1 \wedge \cdots \wedge dx_n) = \sum_{r=1}^{n}(-1)^{r-1}f_r dx_1 \wedge \cdots \wedge \widehat{dx_r} \wedge \cdots \wedge dx_n,</math>
where <math>dx_1\wedge \cdots \wedge \widehat{dx_r} \wedge \cdots \wedge dx_n</math> is the form obtained by omitting <math>dx_r</math> from <math>dx_1 \wedge \cdots \wedge dx_n</math>.
By antisymmetry of forms,
<math display=block>\iota_X \iota_Y \omega = -\iota_Y \iota_X \omega,</math>
and so <math>\iota_X \circ \iota_X = 0.</math> This may be compared to the exterior derivative <math>d,</math> which has the property <math>d \circ d = 0.</math>
The interior product with respect to the commutator of two vector fields <math>X,</math> <math>Y</math> satisfies the identity
<math display="block">\iota_{[X,Y]} = \left[\mathcal{L}_X, \iota_Y\right] = \left[\iota_X, \mathcal{L}_Y\right]. </math>Proof. For any k-form <math>\Omega</math>, <math display="block">\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_{\mathcal L_X Y}\Omega = \iota_{[X,Y]}\Omega</math>and similarly for the other result.
Cartan identity
The interior product relates the exterior derivative and Lie derivative of differential forms by the <span id="Cartan formula">Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula)</span>:
<math display="block">\mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\{ d, \iota_X \right\} \omega.</math>
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 momentum map. The Cartan homotopy formula is named after Élie Cartan.