Interior product

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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 ιXω is sometimes written as Xω.[1]


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 ιXω defined by the property that

for any vector fields X1, ..., Xp−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 called a derivation.


By antisymmetry of forms,

and so . This may be compared to the exterior derivative d, which has the property dd = 0.

The interior product relates the exterior derivative and Lie derivative of differential forms by the Cartan formula (a.k.a. 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 also[edit]


  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


  • 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