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 {\displaystyle \iota _{X}\omega } is sometimes written as ω ⌊ X {\displaystyle \omega \mathbin {\lfloor } X} , which is called the right contraction of ω {\displaystyle \omega } with X.
Definition The interior product is defined to be the contraction of a differential form with a vector field. Thus if X {\displaystyle X} is a vector field on the manifold M , {\displaystyle M,} then
ι X : Ω p ( M ) → Ω p − 1 ( M ) {\displaystyle \iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)}
is the map which sends a p {\displaystyle p} -form ω {\displaystyle \omega } to the ( p − 1 ) {\displaystyle (p-1)} -form ι X ω {\displaystyle \iota _{X}\omega } defined by the property that
( ι X ω ) ( X 1 , … , X p − 1 ) = ω ( X , X 1 , … , X p − 1 ) {\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)}
for any vector fields X 1 , … , X p − 1 . {\displaystyle X_{1},\ldots ,X_{p-1}.}
When ω {\displaystyle \omega } is a scalar field (0-form), ι X ω = 0 {\displaystyle \iota _{X}\omega =0} by convention. The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms α {\displaystyle \alpha }
ι X α = α ( X ) = ⟨ α , X ⟩ , {\displaystyle \displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,}
where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \,\cdot ,\cdot \,\rangle } is the duality pairing between α {\displaystyle \alpha } and the vector X . {\displaystyle X.} Explicitly, if α {\displaystyle \alpha } is a p {\displaystyle p} -form and β {\displaystyle \beta } is a q {\displaystyle q} -form, then
ι X ( α ∧ β ) = ( ι X α ) ∧ β + ( − 1 ) p α ∧ ( ι X β ) . {\displaystyle \iota _{X}(\alpha \wedge \beta )=\left(\iota _{X}\alpha \right)\wedge \beta +(-1)^{p}\alpha \wedge \left(\iota _{X}\beta \right).}
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 , … , x n ) {\displaystyle (x_{1},\ldots ,x_{n})} the vector field X {\displaystyle X} is given by
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