Английская Википедия: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 <math>\iota_X \omega</math> is sometimes written as <math>X \mathbin{\lrcorner} \omega.</math>[1]
Definition
The interior product is defined to be the contraction of a differential form with a vector field. Thus if <math>X</math> is a vector field on the manifold <math>M,</math> then <math display=block>\iota_X : \Omega^p(M) \to \Omega^{p-1}(M)</math> is the map which sends a <math>p</math>-form <math>\omega</math> to the <math>(p - 1)</math>-form <math>\iota_X \omega</math> defined by the property that <math display=block>(\iota_X\omega)\left(X_1, \ldots, X_{p-1}\right) = \omega\left(X, X_1, \ldots, X_{p-1}\right)</math> for any vector fields <math>X_1, \ldots, X_{p-1}.</math>
The interior product is the unique antiderivation of degree −1 on the exterior algebra such that on one-forms <math>\alpha</math> <math display=block>\displaystyle\iota_X \alpha = \alpha(X) = \langle \alpha, X \rangle,</math> where <math>\langle \,\cdot, \cdot\, \rangle</math> is the duality pairing between <math>\alpha</math> and the vector <math>X.</math> Explicitly, if <math>\beta</math> is a <math>p</math>-form and <math>\gamma</math> is a <math>q</math>-form, then <math display=block>\iota_X(\beta \wedge \gamma) = \left(\iota_X\beta\right) \wedge \gamma + (-1)^p \beta \wedge \left(\iota_X\gamma\right).</math> 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 <math>(x_1,...,x_n)</math> the vector field <math>X</math> is given by
<math>X = f_1 \frac{\partial}{\partial x_1} + \cdots + f_n \frac{\partial}{\partial x_n} </math>
then the interior product is given by <math display="block">\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,</math> where <math>dx_1\wedge ...\wedge \widehat{dx_r} \wedge ... \wedge dx_n</math> is the form obtained by omitting <math>dx_r</math> from <math>dx_1 \wedge ...\wedge dx_n</math>.
By antisymmetry of forms, <math display=block>\iota_X \iota_Y \omega = - \iota_Y \iota_X \omega,</math> and so <math>\iota_X \circ \iota_X = 0.</math> This may be compared to the exterior derivative <math>d,</math> which has the property <math>d \circ d = 0.</math>
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): <math display=block>\mathcal L_X\omega = d(\iota_X \omega) + \iota_X d\omega = \left\{ d, \iota_X \right\} \omega.</math>
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 <math>X,</math> <math>Y</math> satisfies the identity <math display=block>\iota_{[X,Y]} = \left[\mathcal{L}_X, \iota_Y\right].</math>
See also
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. Шаблон:Doi
Шаблон:Manifolds Шаблон:Tensors
- ↑ The character ⨼ is U+2A3C INTERIOR PRODUCT in Unicode
- ↑ Tu, Sec 20.5.
- ↑ There is another formula called "Cartan formula". See Steenrod algebra.
- ↑ Шаблон:Citation
