nLab
invariant differential form

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Invariant differential forms and vector fields

Definition

Let MM be a differential manifold with differentiable left action of Lie group GG, G×MMG\times M\to M (respectively right action M×GGM\times G\to G). For example, the multiplication map of GG on itself. Then we define the left translations L g:mgmL_g : m\mapsto g m (resp. right translations R g:mmgR_g: m\mapsto m g) for every gGg\in G, which are both diffeomorphisms of MM.

A differential form on a Lie group ωΩ 1(G)\omega \in \Omega^1(G) is called left invariant if for every gGg \in G it is invariant under the pullback by the translation L gL_g

(L g) *ω=ω(L_g)^* \omega = \omega.

Analogously a form is right invariant if it is invariant under the pullback by right translations R gR_g. For a vector field XX one instead typically defines the invariance via the pushforward (TL g)X=(L g) *X(T L_g) X = (L_g)_* X. Regarding that L gL_g and T gT_g are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.

References

page 89 (20 of 49) at

  • MIT course on Lie groups (pdf 2)
  • Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces
  • F. Bruhat, Lectures on Lie groups and representations of locally compact groups, notes by S. Ramanan, TATA Bombay 1958, 1968, pdf

Revised on May 27, 2013 22:04:37 by Urs Schreiber (82.113.99.197)