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)
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In differential topology, given a pair , of smooth manifolds, and a pair of parallel smooth functions between them, a smooth homotopy between these is (typically defined to be) a left homotopy by a smooth function, hence a smooth function on the product manifold of with the real line, which restricts to at , for .
More generally, for , differentiable manifolds and and differentiable functions, one may ask for differentiable homotopies, all to any given order(s) of differentiability.
At least if and are closed manifolds, then a smooth homotopy exists between any pair of parallel smooth functions between them as soon as there exists an ordinary (i.e. continuous) left homotopy between their underlying continuous functions.
More generally, if are -fold differentiable, then an ordinary continuous homotopy between them implies an -fold differentiable homotopy.
(Pontrjagin 55, Thm. 8, p. 41)
Created on February 3, 2021 at 10:43:11. See the history of this page for a list of all contributions to it.