Interior product

Summary

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 is sometimes written as [1]

DefinitionEdit

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  

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.

PropertiesEdit

By antisymmetry of forms,

 
and so   This may be compared to the exterior derivative   which has the property  

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     satisfies the identity

 

See alsoEdit

NotesEdit

  1. ^ The character ⨼ is U+2A3C in Unicode
  2. ^ Tu, Sec 20.5.
  3. ^ There is another formula called "Cartan formula". See Steenrod algebra.
  4. ^ Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25

ReferencesEdit

  • 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