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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
The (strong) Whitney embedding theorem states that every smooth manifold (Hausdorff and sigma-compact) of dimension $n$ has an embedding of smooth manifolds in the Euclidean space of dimension of dimension $2n$.
Notice that it is easy to see that every smooth manifold embeds into the Eucludean space of some dimension (this prop.). The force of Whitney’s strong embedding theorem is to find the lowest dimension that still works in general.
Named after Hassler Whitney.
See also
Wikipedia, Whitney embedding theorem
Paul Rapoport, Introduction to Immersion, Embeddingand the Whitney Embedding Theorem, 2015 (pdf)
Created on May 6, 2017 at 15:20:08. See the history of this page for a list of all contributions to it.