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)
Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
(Cartan's closed subgroup theorem)
If is a closed subgroup of a (finite dimensional) Lie group, then is a sub-Lie group, hence a smooth submanifold such that its group operations are smooth functions with respect to the submanifold smooth structure.
(due to Cartan 52, see Lee 12, Thm. 20.12)
The statement is originally due to:
Textbook accounts:
Jeffrey M. Lee, Theorem 5.81 in: Manifolds and Differential Geometry, Graduate Studies in Mathematics 107, AMS 2009
John Lee, Theorem 20.12 in: Introduction to Smooth Manifolds, Second Edition, Graduate Texts in Mathematics 218 (2012), Springer (doi:10.1007/978-1-4419-9982-5, book webpage, pdf)
Lecture notes:
See also
Last revised on September 6, 2021 at 12:58:28. See the history of this page for a list of all contributions to it.