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
is sometimes written as
[1]
Definition
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
When
is a scalar field (0-form),
by convention.
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.
Properties
If in local coordinates
the vector field
is given by
then the interior product is given by
where
is the form obtained by omitting
from
.
By antisymmetry of forms,
and so
This may be compared to the exterior derivative
which has the property
The interior product with respect to the commutator of two vector fields
satisfies the identity
Proof. For any k-form
,
and similarly for the other result.
Cartan identity
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):
where the anticommutator was used. 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]
Proof by direct computation [5] Since vector fields are locally integrable, we can always find a local coordinate system
such that the vector field
corresponds to the partial derivative with respect to the first coordinate, i.e.,
.
By linearity of the interior product, exterior derivative, and Lie derivative, it suffices to prove the Cartan's magic formula for monomial
-forms. There are only two cases:
Case 1:
. Direct computation yields:
Case 2:
. Direct computation yields:
See also
- Cap product – Method in algebraic topology
- Inner product – Generalization of the dot product; used to define Hilbert spacesPages displaying short descriptions of redirect targets
- Tensor contraction – Operation in mathematics and physics
Notes
- ^ The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode
- ^ Tu, Sec 20.5.
- ^ There is another formula called "Cartan formula". See Steenrod algebra.
- ^ a b Is "Cartan's magic formula" due to Élie or Henri?, MathOverflow, 2010-09-21, retrieved 2018-06-25
- ^ Elementary Proof of the Cartan Magic Formula, Oleg Zubelevich
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
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