Interior product

Not to be confused withInner product.

Inmathematics,theinterior product(also known asinterior derivative,interior multiplication,inner multiplication,inner derivative,insertion operator,orinner derivation) is adegree−1(anti)derivationon theexterior algebraofdifferential formson asmooth manifold.The interior product, named in opposition to theexterior product,should not be confused with aninner product.The interior product$\iota _{X}\ Omega$is sometimes written as$X\mathbin {\lrcorner } \ Omega.$

Definition

The interior product is defined to be thecontractionof adifferential formwith avector field.Thus if$X$is a vector field on themanifold$M,$then $$\iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)$$ is themapwhich sends a$p$-form$\ Omega$to the$(p-1)$-form$\iota _{X}\ Omega$defined by the property that $$(\iota _{X}\ Omega )\left(X_{1},\ldots,X_{p-1}\right)=\ Omega \left(X,X_{1},\ldots,X_{p-1}\right)$$ for any vector fields$X_{1},\ldots,X_{p-1}.$

The interior product is the uniqueantiderivationof degree −1 on theexterior algebrasuch that on one-forms$\ Alpha$ $$\displaystyle \iota _{X}\ Alpha =\ Alpha (X)=\langle \ Alpha,X\rangle,$$ where$\langle \,\cdot,\cdot \,\rangle$is theduality pairingbetween$\ Alpha$and the vector$X.$Explicitly, if$\beta$is a$p$-form and$\gamma$is a$q$-form, then $$\iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).$$ The above relation says that the interior product obeys a gradedLeibniz rule.An operation satisfying linearity and a Leibniz rule is called a derivation.

Properties

If in local coordinates$(x_{1},...,x_{n})$the vector field$X$is given by

$X=f_{1}{\frac {\partial }{\partial x_{1}}}+\cdots +f_{n}{\frac {\partial }{\partial x_{n}}}$

then the interior product is given by $$\iota _{X}(dx_{1}\wedge...\wedge dx_{n})=\sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}\wedge...\wedge {\widehat {dx_{r}}}\wedge...\wedge dx_{n},$$ where$dx_{1}\wedge...\wedge {\widehat {dx_{r}}}\wedge...\wedge dx_{n}$is the form obtained by omitting$dx_{r}$from$dx_{1}\wedge...\wedge dx_{n}$.

By antisymmetry of forms, $$\iota _{X}\iota _{Y}\ Omega =-\iota _{Y}\iota _{X}\ Omega,$$ and so$\iota _{X}\circ \iota _{X}=0.$This may be compared to theexterior derivative$d,$which has the property$d\circ d=0.$

The interior product relates theexterior derivativeandLie derivativeof differential forms by theCartan formula(also known as theCartan identity,Cartan homotopy formulaorCartan magic formula): $${\mathcal {L}}_{X}\ Omega =d(\iota _{X}\ Omega )+\iota _{X}d\ Omega =\left\{d,\iota _{X}\right\}\ Omega.$$

where theanticommutatorwas used. This identity defines a duality between the exterior and interior derivatives. Cartan's identity is important insymplectic geometryandgeneral relativity:seemoment map.The Cartan homotopy formula is named afterÉlie Cartan.

The interior product with respect to the commutator of two vector fields$X,$$Y$satisfies the identity $$\iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right].$$

See also

• Cap product– Method in algebraic topology
• Inner product– Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
• Tensor contraction– Operation in mathematics and physics

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