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, which is called the right contraction with X.

Definition

The interior product is defined to be the contraction of a differential form with a vector field is a vector field on the manifold is the map

by convention.

The interior product is the unique antiderivation of degree 1 on the exterior algebra is the duality pairing 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

is given by

.

This may be compared to the exterior derivative

Proof.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 or Cartan magic formula)

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. The Cartan homotopy formula is named after Élie Cartan.

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 is defined as

denotes the Hodge dual. Similarly, the left interior product, or left contraction is defined as

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

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

This is the exact analog of the Leibniz product rule above.

See also

Notes

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.