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 is sometimes written as , which is called the right contraction of with X.
The interior product is defined to be the contraction of a differential form with a vector field. Thus if is a vector field on the manifold then
is the map which sends a -form to the -form defined by the property that
for any vector fields
When is a scalar field (0-form), by convention.
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms
where is the duality pairing between and the vector Explicitly, if is a -form and is a -form, then
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.
If in local coordinates the vector field is given by
then the interior product is given by
where is the form obtained by omitting from .
By antisymmetry of forms,
and so This may be compared to the exterior derivative which has the property
The interior product with respect to the commutator of two vector fields satisfies the identity Proof. For any k-form , and similarly for the other result.
The interior product relates the exterior derivative and Lie derivative of differential forms by the <span id="Cartan formula">Cartan formula (also known as the Cartan identity, Cartan homotopy formula or Cartan magic formula)</span>:
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 moment map. The Cartan homotopy formula is named after ÃÂlie Cartan.
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
where 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
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
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 given for the operator above.