Interior product

In mathematics, the interior product or interior derivative 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, is also called interior or inner multiplication, or the inner derivative or derivation, but should not be confused with an inner product. The interior product ι_Xω is sometimes written as X ⨼ ω.

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_Xω defined by the property that

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

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.

By antisymmetry of forms,

and so ${\displaystyle \iota _{X}^{2}=0}$. This may be compared to the exterior derivative d, which has the property d² = 0. The interior product relates the exterior derivative and Lie derivative of differential forms by Cartan's identity:

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