Interior product

 ( Differential Forms ) 

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 −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 $\backslash iota\_X\; \backslash omega$ is sometimes written as $X\; \backslash mathbin\{\backslash lrcorner\}\; \backslash omega.$The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode

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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 $$\backslash iota\_X\; :\; \backslash Omega^p(M)\; \backslash to\; \backslash Omega^\{p-1\}(M)$$ is the map which sends a $p$-form $\backslash omega$ to the $(p\; -\; 1)$-form $\backslash iota\_X\; \backslash omega$ defined by the property that $$(\backslash iota\_X\backslash omega)\backslash left(X\_1,\; \backslash ldots,\; X\_\{p-1\}\backslash right)\; =\; \backslash omega\backslash left(X,\; X\_1,\; \backslash ldots,\; X\_\{p-1\}\backslash right)$$ for any vector fields $X\_1,\; \backslash ldots,\; X\_\{p-1\}.$

When $\backslash omega$ is a scalar field (0-form), $\backslash iota\_X\; \backslash omega\; =\; 0$ by convention.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms $\backslash alpha$ $$\backslash displaystyle\backslash iota\_X\; \backslash alpha\; =\; \backslash alpha(X)\; =\; \backslash langle\; \backslash alpha,\; X\; \backslash rangle,$$ where $\backslash langle\; \backslash ,\backslash cdot,\; \backslash cdot\backslash ,\; \backslash rangle$ is the duality pairing between $\backslash alpha$ and the vector $X.$ Explicitly, if $\backslash beta$ is a $p$-form and $\backslash gamma$ is a $q$-form, then $$\backslash iota\_X(\backslash beta\; \backslash wedge\; \backslash gamma)\; =\; \backslash left(\backslash iota\_X\backslash beta\backslash right)\; \backslash wedge\; \backslash gamma\; +\; (-1)^p\; \backslash beta\; \backslash wedge\; \backslash left(\backslash iota\_X\backslash gamma\backslash right).$$ The above relation says that the interior product obeys a graded Product rule. An operation satisfying linearity and a Leibniz rule is called a derivation.

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Properties If in local coordinates $(x\_1,...,x\_n)$ the vector field $X$ is given by

$X\; =\; f\_1\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_1\}\; +\; \backslash cdots\; +\; f\_n\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\_n\}$

then the interior product is given by $$\backslash iota\_X\; (dx\_1\; \backslash wedge\; ...\backslash wedge\; dx\_n)\; =\; \backslash sum\_\{r=1\}^\{n\}(-1)^\{r-1\}f\_r\; dx\_1\; \backslash wedge\; ...\backslash wedge\; \backslash widehat\{dx\_r\}\; \backslash wedge\; ...\; \backslash wedge\; dx\_n,$$ where $dx\_1\backslash wedge\; ...\backslash wedge\; \backslash widehat\{dx\_r\}\; \backslash wedge\; ...\; \backslash wedge\; dx\_n$ is the form obtained by omitting $dx\_r$ from $dx\_1\; \backslash wedge\; ...\backslash wedge\; dx\_n$.

By antisymmetry of forms, $$\backslash iota\_X\; \backslash iota\_Y\; \backslash omega\; =\; -\; \backslash iota\_Y\; \backslash iota\_X\; \backslash omega,$$ and so $\backslash iota\_X\; \backslash circ\; \backslash iota\_X\; =\; 0.$ This may be compared to the exterior derivative $d,$ which has the property $d\; \backslash circ\; d\; =\; 0.$

The interior product with respect to the commutator of two vector fields $X,$ $Y$ satisfies the identity $$\backslash iota\_\{X,Y\}\; =\; \backslash left\backslash mathcal\{L\}\_X,\; =\; \backslash left\backslash iota\_X,.$$ Proof. For any k-form $\backslash Omega$, $$\backslash mathcal\; L\_X(\backslash iota\_Y\; \backslash Omega)\; -\; \backslash iota\_Y\; (\backslash mathcal\; L\_X\backslash Omega)\; =\; (\backslash mathcal\; L\_X\backslash Omega)(Y,\; -)\; +\; \backslash Omega(\backslash mathcal\; L\_X\; Y,\; -)\; -\; (\backslash mathcal\; L\_X\; \backslash Omega)(Y\; ,\; -)\; =\; \backslash iota\_\{\backslash mathcal\; L\_X\; Y\}\backslash Omega\; =\; \backslash iota\_\{X,Y\}\backslash Omega$$and similarly for the other result.

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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 formulaTu, Sec 20.5. or Cartan magic formula): $$\backslash mathcal\; L\_X\backslash omega\; =\; d(\backslash iota\_X\; \backslash omega)\; +\; \backslash iota\_X\; d\backslash omega\; =\; \backslash left\backslash \{\; d,\; \backslash iota\_X\; \backslash right\backslash \}\; \backslash omega.$$

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.

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See also

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Notes

• Theodore Frankel, The Geometry of Physics: An Introduction; Cambridge University Press, 3rd ed. 2011
• Loring W. Tu, An Introduction to Manifolds, 2e, Springer. 2011.

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