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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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 .
A differential form on a Lie group is called left invariant if for every it is invariant under the pullback by the translation
.
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.
Last revised on July 8, 2018 at 05:48:37. See the history of this page for a list of all contributions to it.