Poincaré 1-form

Definition 1.

Suppose M is a manifold, and TM is its cotangent bundleMathworldPlanetmath. Then the , αΩ1(TM), is locally defined as


where xi,yi are canonical local coordinates for TM.

Let us show that the Poincaré 1-form is globally defined. That is, α has the same expression in all local coordinates. Suppose xi,x~i are overlapping coordinates for M. Then we have overlapping local coordinates (xi,yi), (x~i,y~i) for TM with the transformation rule



i=1ny~idx~i = i=1ny~ix~ixkdxk
= i=1nx~jxiyjx~ixkdxk
= k=1nykdxk.


  1. 1.

    The Poincaré 1-form play a crucial role in symplectic geometry. The form dα is the canonical symplectic formMathworldPlanetmath for TM.

  2. 2.

    Suppose π:TMM is the canonical projection. Then


    which is an alternative definition of α without local coordinates.

  3. 3.

    The restriction of this form to the unit cotangent bundle, is a contact form.

Title Poincaré 1-form
Canonical name Poincare1form
Date of creation 2013-03-22 14:45:44
Last modified on 2013-03-22 14:45:44
Owner matte (1858)
Last modified by matte (1858)
Numerical id 7
Author matte (1858)
Entry type Definition
Classification msc 58A32
Synonym Liouville one-form