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)
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
Given a smooth manifold and a vector field on it, one defines a series of operators on spaces of differential forms, of functions, of vector fields and multivector fields. For functions (derivative of along an integral curve of ); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.
For vector fields . If is a differential form on , the Lie derivative of along is the linearization of the pullback of along the flow induced by
Denote by be the graded derivation which is the contraction with a vector field . By Cartan's homotopy formula,
Cartan introduced Lie derivatives of differential forms and derived Cartan's magic formula in
Extension to arbitrary tensor fields was given in
The term “Lie derivative” (Liesche Ableitung) is due to van Dantzig, who also suggested a definition using the flow of a vector field:
An introduction in the context of synthetic differential geometry is in
A gentle elementary introduction for mathematical physicists
.
Last revised on April 4, 2021 at 04:21:16. See the history of this page for a list of all contributions to it.