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)
A classical text regarded as envisioning the modern notion of (differentiable, smooth) manifolds and of what came to be called Riemannian geometry.
Über die Hypothesen, welche der Geometrie zu Grunde liegen
talk before the Göttingen Faculty (including Gauss, Dedekind & Weber)
June 10, 1854
Reprinted with historical commentary as:
Jürgen Jost (ed.)
Klassische Texte der Wissenschaft
Springer (2013)
English translation by William Clifford:
On the hypotheses which underlie geometry
Nature VIII (1873) 183-184
Reprinted with historical commentary in:
Jürgen Jost (ed.)
On the Hypotheses Which Lie at the Bases of Geometry
Classic Texts in the Sciences
Springer (2016)
Quotes:
On the potential breakdown of differential-geometric space at microscopic distances, reconsidered much later in the context of quantum gravity:
[§III.3] Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena.
Last revised on January 19, 2024 at 09:38:13. See the history of this page for a list of all contributions to it.