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.
See most textbooks on on Lie theory, e.g.:
Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]
François Bruhat (notes by S. Ramanan): Lectures on Lie groups and representations of locally compact groups, Tata Institute Bombay (1958, 1968) [pdf, pdf]
See also:
page 89 (20 of 49) at MIT course on Lie groups (pdf 2)
MathOverflow: Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids [MO:q/178528]
Last revised on July 11, 2024 at 10:52:58. See the history of this page for a list of all contributions to it.