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2014-01-26T02:16:18Z

65.92.5.161:

{{distinguish|Inner product}}
In [[mathematics]], the '''interior product''' is a [[graded algebra|degree]] −1 [[antiderivation]] on the [[exterior algebra]] of [[differential form]]s on a [[smooth manifold]]. The interior product, named in opposition to the [[exterior product]], is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an [[inner product]]. The interior product ''ι''''X''''ω'' is sometimes written as ''X'' {{Unicode|⨼}} ''ω''; this character is U+2A3C in [[Unicode]].

==Definition==

The interior product is defined to be the [[tensor contraction|contraction]] of a [[differential form]] with a [[vector field]]. Thus if ''X'' is a vector field on the [[manifold]] ''M'', then
:<math>\iota_X\colon \Omega^p(M) \to \Omega^{p-1}(M)</math>
is the [[Map (mathematics)|map]] which sends a ''p''-form ''ω'' to the (''p''−1)-form ''ι''''X''''ω'' defined by the property that
:<math>( \iota_X\omega )(X_1,\ldots,X_{p-1})=\omega(X,X_1,\ldots,X_{p-1})</math>
for any vector fields ''X''1,..., ''X''''p''−1.

The interior product is the unique [[derivation (algebra)|antiderivation]] of degree −1 on the [[exterior algebra]] such that on one-forms ''α''
:<math>\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha,X \rangle</math>,
the duality pairing between ''α'' and the vector ''X''. Explicitly, if ''β'' is a ''p''-form and γ is a ''q''-form, then
:<math> \iota_X(\beta\wedge\gamma) = (\iota_X\beta)\wedge\gamma+(-1)^p\beta\wedge(\iota_X\gamma). </math>
The above relation says that the interior product obeys a graded [[Product rule|Leibniz rule]]. An operation equipped with linearity and a Leibniz rule is often called a derivative. The interior product is also known as the interior derivative.

==Properties==
By antisymmetry of forms,

:<math> \iota_X \iota_Y \omega = - \iota_Y \iota_X^{ } \omega </math>

and so <math> \iota_X^2 = 0 </math>. This may be compared to the [[exterior derivative]] ''d'' which has the property ''d''2 = 0. The interior product relates the [[exterior derivative]] and [[Lie derivative]] of differential forms by '''''Cartan's identity''''':
:<math> \mathcal L_X\omega = \mathrm d (\iota_X \omega) + \iota_X \mathrm d\omega. </math>
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 interior product with respect to the commutator of two vector fields <math>X,Y</math> satisfies the identity

:<math> \iota_{[X,Y]}=\mathcal L_X \iota_Y-\iota_Y \mathcal L_X. </math>

==See also==
* [[Inner product]]
* [[Tensor contraction]]

{{DEFAULTSORT:Interior Product}}
[[Category:Differential forms]]
[[Category:Multilinear algebra]]

{{differential-geometry-stub}}

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