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
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
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
A smooth isotopy is an isotopy that varies smoothly, hence a isotopy that, as a left homotopy, is a smooth homotopy.
In knot theory, one typically does not want to identify knots $S^1 \to S^3$ by plain isotopy, as that makes all tame knots? be equivalent. A common fix is to use ambient isotopy instead. But one may also use smooth isotopy. (see e.g. Greene 13 or MO discussion here).
Let $\Sigma$ and $X$ be smooth manifolds, and let
be two embeddings of smooth manifolds. Then a smooth isotopy between them is a smooth homotopy between them via embeddings: a smooth function
such that
and such that for each $t \in [0,1]$,
is an embedding of smooth manifolds.
(e.g. Greene 13, Def. 1.7)
Last revised on February 3, 2021 at 15:56:58. See the history of this page for a list of all contributions to it.