“Interior product”的意思、由来-开放百科全书

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Interior product

释义

Definition
Properties
See also
Notes
References

In mathematics, the interior product ({{aka}} 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 ι_Xω is sometimes written as {{nowrap|X ⨼ ω}}.^[1]

Definition

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 {{nowrap|⟨ , ⟩}} 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.

Properties

By antisymmetry of forms,

and so . This may be compared to the exterior derivative d, which has the property {{nowrap|1=d ∘ d = 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

Notes

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. ^{{citation | title=Is "Cartan's magic formula" due to Élie or Henri? | url=https://mathoverflow.net/q/39540 | accessdate=2018-06-25 | date=2010-09-21 |publisher=mathoverflow}}

References

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}}

2 : Differential forms|Multilinear algebra

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Definition

Properties

See also

Notes

References