Definition Let ω be a smooth k-form on a smooth manifoldMathworldPlanetmath M, and let ξ be a smooth vector field on M. The contraction of ω with ξ is the smooth (k-1)-form that maps xM to ωx(ξx,). In other words, ω is point-wise evaluated with ξ in the first slot. We shall denote this (k-1)-form by ιξω. If ω is a 0-form, we set ιξω=0 for all ξ.

Properties Let ω and ξ be as above. Then the following properties hold:

  1. 1.

    For any real number k

  2. 2.

    For vector fields ξ and η

    ιξ+ηω = ιξω+ιηω,
    ιξιηω = -ιηιξω,
    ιξιξω = 0.
  3. 3.

    Contraction is an anti-derivation [1]. If ω1 is a p-form, and ω2 is a q-form, then



  • 1 T. Frankel, Geometry of physics, Cambridge University press, 1997.
Title contraction
Canonical name Contraction
Date of creation 2013-03-22 13:37:28
Last modified on 2013-03-22 13:37:28
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 4
Author mathcam (2727)
Entry type Definition
Classification msc 15A75
Classification msc 58A10