synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A differential form on a Lie group is called left invariant if for every it is invariant under the pullback of differential forms
along the left multiplication action
Analogously a form is right invariant if it is invariant under the pullback by right translations .
More generally, given a differentiable (e.g. smooth) group action of on a differentiable (e.g. smooth) manifold
then a differential form is called invariant if for all
This reduces to the left invariance (1) for and being the left multiplication action of on itself.
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.
Last revised on April 27, 2024 at 10:22:29. See the history of this page for a list of all contributions to it.