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
ι
X
ω
{\displaystyle \iota _{X}\omega }
is sometimes written as
X
⌟
ω
.
{\displaystyle X\mathbin {\lrcorner } \omega .}
[1]
Definition
The interior product is defined to be the contraction of a differential form with a vector field . Thus if
X
{\displaystyle X}
is a vector field on the manifold
M
,
{\displaystyle M,}
then
ι
X
:
Ω
p
(
M
)
→
Ω
p
−
1
(
M
)
{\displaystyle \iota _{X}:\Omega ^{p}(M)\to \Omega ^{p-1}(M)}
is the
map which sends a
p
{\displaystyle p}
-form
ω
{\displaystyle \omega }
to the
(
p
−
1
)
{\displaystyle (p-1)}
-form
ι
X
ω
{\displaystyle \iota _{X}\omega }
defined by the property that
(
ι
X
ω
)
(
X
1
,
…
,
X
p
−
1
)
=
ω
(
X
,
X
1
,
…
,
X
p
−
1
)
{\displaystyle (\iota _{X}\omega )\left(X_{1},\ldots ,X_{p-1}\right)=\omega \left(X,X_{1},\ldots ,X_{p-1}\right)}
for any vector fields
X
1
,
…
,
X
p
−
1
.
{\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
α
{\displaystyle \alpha }
ι
X
α
=
α
(
X
)
=
⟨
α
,
X
⟩
,
{\displaystyle \displaystyle \iota _{X}\alpha =\alpha (X)=\langle \alpha ,X\rangle ,}
where
⟨
⋅
,
⋅
⟩
{\displaystyle \langle \,\cdot ,\cdot \,\rangle }
is the
duality pairing between
α
{\displaystyle \alpha }
and the vector
X
.
{\displaystyle X.}
Explicitly, if
β
{\displaystyle \beta }
is a
p
{\displaystyle p}
-form and
γ
{\displaystyle \gamma }
is a
q
{\displaystyle q}
-form, then
ι
X
(
β
∧
γ
)
=
(
ι
X
β
)
∧
γ
+
(
−
1
)
p
β
∧
(
ι
X
γ
)
.
{\displaystyle \iota _{X}(\beta \wedge \gamma )=\left(\iota _{X}\beta \right)\wedge \gamma +(-1)^{p}\beta \wedge \left(\iota _{X}\gamma \right).}
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
(
x
1
,
.
.
.
,
x
n
)
{\displaystyle (x_{1},...,x_{n})}
the vector field
X
{\displaystyle X}
is given by
X
=
f
1
∂
∂
x
1
+
⋯
+
f
n
∂
∂
x
n
{\displaystyle X=f_{1}{\frac {\partial }{\partial x_{1}+\cdots +f_{n}{\frac {\partial }{\partial x_{n}
then the interior product is given by
ι
X
(
d
x
1
∧
.
.
.
∧
d
x
n
)
=
∑
r
=
1
n
(
−
1
)
r
−
1
f
r
d
x
1
∧
.
.
.
∧
d
x
r
^
∧
.
.
.
∧
d
x
n
,
{\displaystyle \iota _{X}(dx_{1}\wedge ...\wedge dx_{n})=\sum _{r=1}^{n}(-1)^{r-1}f_{r}dx_{1}\wedge ...\wedge {\widehat {dx_{r}\wedge ...\wedge dx_{n},}
where
d
x
1
∧
.
.
.
∧
d
x
r
^
∧
.
.
.
∧
d
x
n
{\displaystyle dx_{1}\wedge ...\wedge {\widehat {dx_{r}\wedge ...\wedge dx_{n}
is the form obtained by omitting
d
x
r
{\displaystyle dx_{r}
from
d
x
1
∧
.
.
.
∧
d
x
n
{\displaystyle dx_{1}\wedge ...\wedge dx_{n}
.
By antisymmetry of forms,
ι
X
ι
Y
ω
=
−
ι
Y
ι
X
ω
,
{\displaystyle \iota _{X}\iota _{Y}\omega =-\iota _{Y}\iota _{X}\omega ,}
and so
ι
X
∘
ι
X
=
0.
{\displaystyle \iota _{X}\circ \iota _{X}=0.}
This may be compared to the
exterior derivative
d
,
{\displaystyle d,}
which has the property
d
∘
d
=
0.
{\displaystyle d\circ 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 ) :
L
X
ω
=
d
(
ι
X
ω
)
+
ι
X
d
ω
=
{
d
,
ι
X
}
ω
.
{\displaystyle {\mathcal {L}_{X}\omega =d(\iota _{X}\omega )+\iota _{X}d\omega =\left\{d,\iota _{X}\right\}\omega .}
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]
The interior product with respect to the commutator of two vector fields
X
,
{\displaystyle X,}
Y
{\displaystyle Y}
satisfies the identity
ι
[
X
,
Y
]
=
[
L
X
,
ι
Y
]
.
{\displaystyle \iota _{[X,Y]}=\left[{\mathcal {L}_{X},\iota _{Y}\right].}
See also
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 – Operation in mathematics and physics
Notes
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