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)
Given a Lie group and a smooth -principal bundle over a base smooth manifold , a differential form on the total space is called basic if it is the pullback of differential forms along the bundle projection of a differential form of the base manifold
Equivalently, if denotes the Lie algebra of , and for we write
for the vector field on which is the derivative of the -action along , then differential form is basic precisely if
it is annihilated by the contraction with
it is annihilated by the Lie derivative along :
(where the first equality holds generally by Cartan's magic formula, we are displaying it just for emphasis)
for all .
In this form the definition of basic forms makes sense more generally whenever a Cartan calculus is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil mdoel for equivariant de Rham cohomology.
Created on June 27, 2019 at 16:24:53. See the history of this page for a list of all contributions to it.