# Interior product

In mathematics, the **interior product** (aka **interior derivative**, **interior multiplication**, **inner multiplication**, **inner derivative**, or **inner derivation**) is a degree −1 antiderivation 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 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

is the map which sends a *p*-form *ω* to the (*p*−1)-form *ι*_{X}*ω* defined by the property that

for any vector fields *X*_{1}, ..., *X*_{p−1}.

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

- ,

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

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

## Properties

By antisymmetry of forms,

and so
. This may be compared to the exterior derivative *d*, which has the property *d*^{2} = 0.

The interior product relates the exterior derivative and Lie derivative of differential forms by **Cartan formula** (a.k.a. **Cartan identity**, **Cartan homotopy formula**^{[2]} or **Cartan magic formula**):

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.) Cartan homotopy formula is named after Élie Cartan.^{[3]}

The interior product with respect to the commutator of two vector fields , satisfies the identity

## See also

## Notes

- ↑ The character ⨼ is U+2A3C in Unicode
- ↑ Tu, Sec 20.5.
- ↑
*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