nLab
invariant differential form

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

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 January 10, 2017 16:22:39 by Urs Schreiber (87.191.41.20)