Interior product

In mathematics, the interior product (also known as 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.[1]

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

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

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

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

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

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

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

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

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,

${\displaystyle \iota _{X}\iota _{Y}\omega =-\iota _{Y}\iota _{X}\omega }$

and so ${\displaystyle \iota _{X}\circ \iota _{X}=0}$. 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[2] or Cartan magic formula):

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

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.[3] The Cartan homotopy formula is named after Élie Cartan.[4]

The interior product with respect to the commutator of two vector fields ${\displaystyle X}$, ${\displaystyle Y}$ satisfies the identity

${\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}}_{X},\iota _{Y}\right].}$

See also

• Cap product
• Inner product
• Tensor contraction

Notes

1. The character ⨼ is U+2A3C in Unicode
2. Tu, Sec 20.5.
3. There is another formula called "Cartan formula". See Steenrod algebra.
4. Is "Cartan's magic formula" due to Élie or Henri?, mathoverflow, 2010-09-21, retrieved 2018-06-25

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