Knowlpedia

Interior product (contraction) ι_X

Insertion of a vector field into a differential form, producing a form of one lower degree.

Let MM be a and let XX be a on MM. For a ω∈Ωk(M)\omega\in\Omega^k(M), the interior product (or contraction) of ω\omega by XX is the (k−1)(k-1)-form ιXω∈Ωk−1(M)\iota_X\omega\in\Omega^{k-1}(M) defined by

(ιXω)p(v1,…,vk−1)  :=  ωp(Xp,v1,…,vk−1),p∈M,  vi∈TpM.(\iota_X\omega)_p(v_1,\dots,v_{k-1}) \;:=\; \omega_p(X_p,v_1,\dots,v_{k-1}), \qquad p\in M,\; v_i\in T_pM.

By convention, if k=0k=0 then ιXω:=0\iota_X\omega:=0.

The contraction satisfies the following standard identities.

  • C∞(M)C^\infty(M)-linearity in the vector field: for f∈C∞(M)f\in C^\infty(M),
    ιfXω=f ιXω.\iota_{fX}\omega = f\,\iota_X\omega.
  • Graded derivation rule for the : if α∈Ωp(M)\alpha\in\Omega^p(M) and β∈Ωq(M)\beta\in\Omega^q(M), then
    ιX(α∧β)=(ιXα)∧β+(−1)p α∧(ιXβ).\iota_X(\alpha\wedge\beta) = (\iota_X\alpha)\wedge\beta + (-1)^p\,\alpha\wedge(\iota_X\beta).
  • Cartan’s formula (interaction with differentiation): the of forms along XX is given by
    LXω=d(ιXω)+ιX(dω),\mathcal{L}_X\omega = d(\iota_X\omega)+\iota_X(d\omega),
    where dd is the .
Examples
  1. If α∈Ω1(M)\alpha\in\Omega^1(M) is a 11-form, then ιXα\iota_X\alpha is the smooth function α(X)∈C∞(M)\alpha(X)\in C^\infty(M) obtained by evaluating α\alpha on the vector field XX.
  1. On M=R2M=\mathbb{R}^2 with coordinates (x,y)(x,y), let ω=dx∧dy\omega=dx\wedge dy and X=∂∂xX=\frac{\partial}{\partial x}. Then
    ιXω=dy.\iota_X\omega = dy.
  1. On M=R3M=\mathbb{R}^3 with coordinates (x,y,z)(x,y,z), let ω=dx∧dy∧dz\omega=dx\wedge dy\wedge dz and X=∂∂x+2∂∂yX=\frac{\partial}{\partial x}+2\frac{\partial}{\partial y}. Then
    ιXω=dy∧dz−2 dx∧dz.\iota_X\omega = dy\wedge dz - 2\,dx\wedge dz.