invariant differential form

Invariant differential forms and vector fields


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \mathrm{R}\!\!\mathrm{h} & \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 }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Invariant differential forms and vector fields


Let MM be a differential manifold with differentiable left action of Lie group GG, G×MMG\times M\to M (respectively right action M×GMM\times G\to M). 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.


Last revised on July 8, 2018 at 05:48:37. See the history of this page for a list of all contributions to it.