View source for Interior product

{{Short description|Mapping from p forms to p-1 forms}}
{{distinguish|Inner product}}

In [[mathematics]], the '''interior product''' (also known as '''interior derivative''', '''interior multiplication''', '''inner multiplication''', '''inner derivative''', '''insertion operator''', or '''inner derivation''') is a [[Graded algebra|degree]] &minus;1 [[Derivation (differential algebra)|(anti)derivation]] on the [[exterior algebra]] of [[differential form]]s 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>X \mathbin{\lrcorner} \omega.</math><ref>The character ⨼ is U+2A3C INTERIOR PRODUCT in [[Unicode]]</ref>

==Definition==

The interior product is defined to be the [[Tensor contraction|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 (mathematics)|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 [[Derivation (algebra)|antiderivation]] of degree &minus;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>\beta</math> is a <math>p</math>-form and <math>\gamma</math> is a <math>q</math>-form, then
<math display="block">\iota_X(\beta \wedge \gamma) = \left(\iota_X\beta\right) \wedge \gamma + (-1)^p \beta \wedge \left(\iota_X\gamma\right).</math>
The above relation says that the interior product obeys a graded [[Product rule|Leibniz rule]]. An operation satisfying linearity and a Leibniz rule is called a derivation.

==Properties==

If in local coordinates <math>(x_1,...,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 ...\wedge dx_n) = \sum_{r=1}^{n}(-1)^{r-1}f_r dx_1 \wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n,</math>
where <math>dx_1\wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n</math> is the form obtained by omitting <math>dx_r</math> from <math>dx_1 \wedge ...\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'''<ref>Tu, Sec 20.5.</ref> 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 [[moment map]].<ref>There is another formula called "Cartan formula". See [[Steenrod algebra]].</ref> The Cartan homotopy formula is named after [[Élie Cartan]].<ref name=":0">{{citation |title=Is "Cartan's magic formula" due to Élie or Henri? |date=2010-09-21 |url=https://mathoverflow.net/q/39540 |access-date=2018-06-25 |publisher=[[MathOverflow]]}}</ref>

{{Math proof|title=Proof by direct computation <ref>''[https://web.ma.utexas.edu/mp_arc/c/20/20-25.pdf Elementary Proof of the Cartan Magic Formula]'', Oleg Zubelevich</ref> |proof=
Since vector fields are locally integrable, we can always find a local coordinate system <math>(\xi^1, \dots, \xi^n)</math> such that the vector field <math>X</math> corresponds to the partial derivative with respect to the first coordinate, i.e., <math>X = \partial_1</math>.

By linearity of the interior product, exterior derivative, and Lie derivative, it suffices to prove the Cartan's magic formula for monomial <math>k</math>-forms. There are only two cases:

Case 1:  <math>\alpha = a \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k</math>. Direct computation yields:<math display="block">
\begin{aligned}
\iota_X \alpha &= a \, d\xi^2 \wedge \dots \wedge d\xi^k, \\
d(\iota_X \alpha) &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k + \sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\
d\alpha &= \sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\
\iota_X(d\alpha) &= -\sum_{i=k+1}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^k, \\
L_X\alpha &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^k.
\end{aligned}
</math>

Case 2:  <math>\alpha = a \, d\xi^2 \wedge d\xi^3 \wedge \dots \wedge d\xi^{k+1} </math>. Direct computation yields:<math display="block">
\begin{aligned}
\iota_X \alpha &= 0, \\
d\alpha &= (\partial_1 a) \, d\xi^1 \wedge d\xi^2 \wedge \dots \wedge d\xi^{k+1} + \sum_{i=k+2}^n (\partial_i a) \, d\xi^i \wedge d\xi^2 \wedge \dots \wedge d\xi^{k+1}, \\
\iota_X(d\alpha) &= (\partial_1 a) \, d\xi^2 \wedge \dots \wedge d\xi^{k+1}, \\
L_X\alpha &= (\partial_1 a) \, d\xi^2 \wedge \dots \wedge d\xi^{k+1}.
\end{aligned}
</math>
}}

{{Math proof|title=Proof by abstract algebra, credited to [[Shiing-Shen Chern]]<ref name=":0" /> |proof=
The exterior derivative <math>d</math> is an anti-derivation on the exterior algebra. Similarly, the interior product <math>\iota_X</math> with a vector field <math>X</math> is also an anti-derivation. On the other hand, the Lie derivative <math>L_X</math> is a derivation.

The anti-commutator of two anti-derivations is a derivation.

To show that two derivations on the exterior algebra are equal, it suffices to show that they agree on a set of generators. Locally, the exterior algebra is generated by 0-forms (smooth functions <math>f</math>) and their differentials, exact 1-forms (<math>df</math>). Verify Cartan's magic formula on these two cases. 
}}

==See also==

* {{annotated link|Cap product}}
* {{annotated link|Inner product}}
* {{annotated link|Tensor contraction}}

==Notes==

{{reflist}}

==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}}

{{Manifolds}}
{{Tensors}}

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