Let be a differential manifold with differentiable left action of Lie group , (respectively right action ). For example, the multiplication map of on itself. Then we define the left translations (resp. right translations ) for every , which are both diffeomorphisms of .
Analogously a form is right invariant if it is invariant under the pullback by right translations . For a vector field one instead typically defines the invariance via the pushforward . Regarding that and are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.
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