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 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 $ι_{X} ω$ is sometimes written, by geometers such as Michael Spivak, as $X ⌟ ω .$ ^[1] It may also be denoted $ω ⌊ X$ , which is called the right contraction of $ω$ with X.

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 $ι_{X} : Ω^{p} (M) \to Ω^{p - 1} (M)$ is the map which sends a $p$ -form $ω$ to the $(p - 1)$ -form $ι_{X} ω$ defined by the property that $(ι_{X} ω) (X_{1}, \dots, X_{p - 1}) = ω (X, X_{1}, \dots, X_{p - 1})$ for any vector fields $X_{1}, \dots, X_{p - 1} .$

When $ω$ is a scalar field (0-form), $ι_{X} ω = 0$ by convention.

The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms $α$ $ι_{X} α = α (X) = ⟨ α, X ⟩,$ where $⟨ \cdot, \cdot ⟩$ is the duality pairing between $α$ and the vector $X .$ Explicitly, if $α$ is a $p$ -form and $β$ is a $q$ -form, then $ι_{X} (α \land β) = (ι_{X} α) \land β + (- 1)^{p} α \land (ι_{X} β) .$ 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

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

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

then the interior product is given by $ι_{X} (d x_{1} \land \dots \land d x_{n}) = \sum_{r = 1}^{n} (- 1)^{r - 1} f_{r} d x_{1} \land \dots \land \hat{d x_{r}} \land \dots \land d x_{n},$ where $d x_{1} \land \dots \land \hat{d x_{r}} \land \dots \land d x_{n}$ is the form obtained by omitting $d x_{r}$ from $d x_{1} \land \dots \land d x_{n}$ .

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

The interior product with respect to the commutator of two vector fields $X,$ $Y$ satisfies the identity $ι_{[X, Y]} = [ℒ_{X}, ι_{Y}] = [ι_{X}, ℒ_{Y}] .$ Proof. For any k-form $Ω$ , $ℒ_{X} (ι_{Y} Ω) - ι_{Y} (ℒ_{X} Ω) = (ℒ_{X} Ω) (Y, -) + Ω (ℒ_{X} Y, -) - (ℒ_{X} Ω) (Y, -) = ι_{ℒ_{X} Y} Ω = ι_{[X, Y]} Ω$ and similarly for the other result.

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 formula^[2] or Cartan magic formula): $ℒ_{X} ω = d (ι_{X} ω) + ι_{X} d ω = {d, ι_{X}} ω .$

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

Proof by direct computation^[5]

Since vector fields are locally integrable, we can always find a local coordinate system $(ξ^{1}, \dots, ξ^{n})$ such that the vector field $X$ corresponds to the partial derivative with respect to the first coordinate, i.e., $X = \partial_{1}$ . (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 $k$ -forms. There are only two cases:

Case 1: $α = a d ξ^{1} \land d ξ^{2} \land \dots \land d ξ^{k}$ . Direct computation yields: $\begin{matrix} ι_{X} α & = a d ξ^{2} \land \dots \land d ξ^{k}, \\ d (ι_{X} α) & = (\partial_{1} a) d ξ^{1} \land d ξ^{2} \land \dots \land d ξ^{k} + \sum_{i = k + 1}^{n} (\partial_{i} a) d ξ^{i} \land d ξ^{2} \land \dots \land d ξ^{k}, \\ d α & = \sum_{i = k + 1}^{n} (\partial_{i} a) d ξ^{i} \land d ξ^{1} \land d ξ^{2} \land \dots \land d ξ^{k}, \\ ι_{X} (d α) & = - \sum_{i = k + 1}^{n} (\partial_{i} a) d ξ^{i} \land d ξ^{2} \land \dots \land d ξ^{k}, \\ L_{X} α & = (\partial_{1} a) d ξ^{1} \land d ξ^{2} \land \dots \land d ξ^{k} . \end{matrix}$

Case 2: $α = a d ξ^{2} \land d ξ^{3} \land \dots \land d ξ^{k + 1}$ . Direct computation yields: $\begin{matrix} ι_{X} α & = 0, \\ d α & = (\partial_{1} a) d ξ^{1} \land d ξ^{2} \land \dots \land d ξ^{k + 1} + \sum_{i = k + 2}^{n} (\partial_{i} a) d ξ^{i} \land d ξ^{2} \land \dots \land d ξ^{k + 1}, \\ ι_{X} (d α) & = (\partial_{1} a) d ξ^{2} \land \dots \land d ξ^{k + 1}, \\ L_{X} α & = (\partial_{1} a) d ξ^{2} \land \dots \land d ξ^{k + 1} . \end{matrix}$

Proof by abstract algebra, credited to Shiing-Shen Chern^[4]

The exterior derivative $d$ is an anti-derivation on the exterior algebra. Similarly, the interior product $ι_{X}$ with a vector field $X$ is also an anti-derivation. On the other hand, the Lie derivative $L_{X}$ 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 $f$ ) and their differentials, exact 1-forms ( $d f$ ). 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, $⌊ : ⋀ V \times ⋀ V \to ⋀ V$ is defined as^[6]

a ⌊ b = a \lor b^{★},

where $\lor$ is the exterior antiproduct (also known as the regressive product), and the superscript $★$ denotes the Hodge dual. Similarly, the left interior product, or left contraction, $⌋$ is defined as

a ⌋ b = a_{★} \lor b,

where the subscript $★$ 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

a ⌊ b = a ⌋ b = a \cdot b .

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

(a \land b) ⌊ X = (a ⌊ X) \land b + (- 1)^{p} a \land (b ⌊ X) .

This is the exact analog of the Leibniz product rule given for the operator $ι_{X}$ above.

Notes

↑ The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode
↑ Tu, Sec 20.5.
↑ There is another formula called "Cartan formula". See Steenrod algebra.
1 2 Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25
↑ Elementary Proof of the Cartan Magic Formula, Oleg Zubelevich
↑ Eric Lengyel (2024). Projective Geometric Algebra Illuminated. Terathon Software. ISBN 979-8-9853582-5-4.

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

[1] The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode

[2] Tu, Sec 20.5.

[3] There is another formula called "Cartan formula". See Steenrod algebra.

[:0-4] 1 2 Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25

[5] Elementary Proof of the Cartan Magic Formula, Oleg Zubelevich

[6] Eric Lengyel (2024). Projective Geometric Algebra Illuminated. Terathon Software. ISBN 979-8-9853582-5-4.

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