Interior product

In [[mathematics]], the '''interior product''' (also known as '''interior derivative''', '''interior multiplication''', '''inner multiplication''', '''inner derivative''', '''insertion operator''', '''contraction''', or '''inner derivation''') is a [[Graded algebra|degree]] −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>\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 [[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 −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 [[Product rule|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'''<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 [[momentum 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>. (See [[Straightening theorem for vector fields]])

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. }}

==In Exterior Algebra==

In the [[exterior algebra]] over a vector space ''V'', the interior product is generalized for arbitrary multivectors ''a'' and ''b''. The ''right interior product'', or ''right contraction'', <math>\textstyle \mathbin{\lfloor} : \bigwedge V \times \bigwedge V \to \bigwedge V</math> is defined as<ref>{{cite book|title=Projective Geometric Algebra Illuminated|isbn=979-8-9853582-5-4 |year=2024 |publisher=Terathon Software |author-link = Eric Lengyel|author=Eric Lengyel}}</ref>

: <math> a \mathbin{\lfloor} b = a \vee b^\bigstar,</math>

where <math>\vee</math> is the exterior antiproduct (also known as the regressive product), and the superscript <math>\bigstar</math> denotes the [[Hodge dual]]. Similarly, the ''left interior product'', or ''left contraction'', <math>\rfloor</math> is defined as

: <math> a \mathbin{\rfloor} b = a_\bigstar \vee b,</math>

where the subscript <math>\bigstar</math> denotes the left version of the Hodge dual.

When ''a'' and ''b'' are homogeneous multivectors with the same grade, then the left and right interior products each reduce to the inner product such that

: <math>a \mathbin{\lfloor} b = a \mathbin{\rfloor} b = a \cdot b.</math>

For a vector ''X'' (which has grade 1), a homogeneous multivector ''a'' having grade ''p'', and an arbitrary multivector ''b'', the right interior product satisfies the rule

: <math>(a \wedge b) \mathbin{\lfloor} X = (a \mathbin{\lfloor} X) \wedge b + (-1)^p a \wedge (b \mathbin{\lfloor} X).</math>

This is the exact analog of the [[Product rule|Leibniz product rule]] given for the operator <math>\iota_X</math> above.

==See also==

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

==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. {{doi|10.1007/978-1-4419-7400-6}}

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