Interior product – Wikipedia

Mapping from p forms to p-1 forms

In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, 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

${displaystyle iota _{X}omega }$

is sometimes written as

${displaystyle Xmathbin {lrcorner } omega .}$

^[1]

Definition[edit]

The interior product is defined to be the contraction of a differential form with a vector field. Thus if

${displaystyle X}$

is a vector field on the manifold

${displaystyle M,}$

then

is the map which sends a

${displaystyle p}$

-form

${displaystyle omega }$

to the

${displaystyle (p-1)}$

-form

${displaystyle iota _{X}omega }$

defined by the property that

for any vector fields

${displaystyle X_{1},ldots ,X_{p-1}.}$

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

${displaystyle alpha }$

where

${displaystyle langle ,cdot ,cdot ,rangle }$

is the duality pairing between

${displaystyle alpha }$

and the vector

${displaystyle X.}$

Explicitly, if

${displaystyle beta }$

is a

${displaystyle p}$

-form and

${displaystyle gamma }$

is a

${displaystyle q}$

-form, then

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

Properties[edit]

If in local coordinates

${displaystyle (x_{1},…,x_{n})}$

the vector field

${displaystyle X}$

is described by functions

${displaystyle f_{1},…,f_{n}}$

, then the interior product is given by

where

${displaystyle dx_{1}wedge …wedge {widehat {dx_{r}}}wedge …wedge dx_{n}}$

is the form obtained by omitting

${displaystyle dx_{r}}$

from

${displaystyle dx_{1}wedge …wedge dx_{n}}$

.

By antisymmetry of forms,

and so

${displaystyle iota _{X}circ iota _{X}=0.}$

This may be compared to the exterior derivative

${displaystyle d,}$

which has the property

${displaystyle dcirc d=0.}$

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^[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.^[3] The Cartan homotopy formula is named after Élie Cartan.^[4]

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

${displaystyle X,}$

${displaystyle Y}$

satisfies the identity

See also[edit]

• Cap product – method of adjoining a chain of with a cochainPages displaying wikidata descriptions as a fallback
• Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
• Tensor contraction – in mathematics and physics, an operation on tensorsPages displaying wikidata descriptions as a fallback

References[edit]

• 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