Interior product

Not to be confused with Inner product.

In mathematics, the interior product (a.k.a. interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −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 ι_Xω is sometimes written as X ⨼ ω.

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

$\iota _{X}\colon \Omega ^{p}(M)\to \Omega ^{p-1}(M)$

is the map which sends a p-form ω to the (p−1)-form ι_Xω defined by the property that

$(\iota _{X}\omega )(X_{1},\ldots ,X_{p-1})=\omega (X,X_{1},\ldots ,X_{p-1})$

for any vector fields X₁, ..., X_p−1.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α

$\displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle$,

where ⟨ , ⟩ is the duality pairing between α and the vector X. Explicitly, if β is a p-form and γ is a q-form, then

$\iota _{X}(\beta \wedge \gamma )=(\iota _{X}\beta )\wedge \gamma +(-1)^{p}\beta \wedge (\iota _{X}\gamma ).$

The above relation says that the interior product obeys a graded Leibniz rule. An operation satisfying linearity and a Leibniz rule is called a derivation.

Properties

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 the exterior derivative d, which has the property d ∘ d = 0.

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (a.k.a. Cartan identity, Cartan homotopy formula or Cartan magic formula):

${\mathcal {L}}_{X}\omega =d(\iota _{X}\omega )+\iota _{X}d\omega =\left\{d,\iota _{X}\right\}\omega .$

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 Cartan homotopy formula is named after Élie 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
• Inner product
• Tensor contraction

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