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)
This entry is about the concept in differential geometry and Lie theory. For the concept in functional analysis see at distribution.
A real distribution on a real smooth manifold $M$ is a real vector subbundle of the tangent bundle $T M$.
A complex distribution is a complex vector subbundle of the complexified tangent bundle $T_{\mathbb{C}}M$ of $M$.
A distribution of hyperplanes is a distribution of codimension $1$ in $T M$; a distribution of complex hyperplanes is a distribution of complex codimension $1$ in $T_{\mathbb{C}} M$.
One class of examples comes from smooth foliations by submanifolds of constant dimension $m\lt n$. Then the tangent vectors at all points to the submanifolds forming the foliation form a distribution of subspaces of dimension $m$. The distributions of that form are said to be integrable.
… say something about the Frobenius theorem …
Discussion in the context of geometric quantization:
Last revised on May 5, 2023 at 05:11:01. See the history of this page for a list of all contributions to it.