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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
Differential geometry is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces…) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional differentiable manifolds modelled locally on topological vector spaces. Techniques of differential calculus can be further stretched to generalized smooth spaces. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, togehter with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called differential topology.
See also generalized smooth space.
Finite-dimensional differential geometry is the geometry modeled on Cartesian spaces and smooth functions between them.
Formally, it is the geometry modeled on the pre-geometry $\mathcal{G} =$CartSp.
This includes a sequence of concepts of generalized smooth spaces:
A smooth manifold (see there for details) is a locally $CartSp$-representable object in the sheaf topos $Sh(CartSp)$.
A diffeological space (see there) is a concrete sheaf in the cohesive topos $Sh(CartSp)$.
a Lie groupoid is a locally representable object in the (2,1)-sheaf (2,1)-topos $Sh_{(2,1)}(CartSp)$;
an ∞-Lie groupoid is an object in the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(CartSp)$.
Similarly, standard models of synthetic differential geometry in higher geometry are modeled on the pre-geometry $\mathcal{G} =$ThCartSp. To wit, the cohesive topos $Sh(ThCartSp)$ is the smooth topos called the Cahiers topos:
an infinitesimal space is a certain object in $ThCartSp$;
an ∞-Lie algebroid is a certain object in the (∞,1)-category of (∞,1)-sheaves $Sh_{(\infty,1)}(ThCartSp)$.
some modern subfields of differential geometry include:
Finsler geometry?,
$\,$
local model | global geometry |
---|---|
Klein geometry | Cartan geometry |
Klein 2-geometry | Cartan 2-geometry |
higher Klein geometry | higher Cartan geometry |
$\,$
$\phantom{A}$(higher) geometry$\phantom{A}$ | $\phantom{A}$site$\phantom{A}$ | $\phantom{A}$sheaf topos$\phantom{A}$ | $\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$ |
---|---|---|---|
$\phantom{A}$discrete geometry$\phantom{A}$ | $\phantom{A}$Point$\phantom{A}$ | $\phantom{A}$Set$\phantom{A}$ | $\phantom{A}$Discrete∞Grpd$\phantom{A}$ |
$\phantom{A}$differential geometry$\phantom{A}$ | $\phantom{A}$CartSp$\phantom{A}$ | $\phantom{A}$SmoothSet$\phantom{A}$ | $\phantom{A}$Smooth∞Grpd$\phantom{A}$ |
$\phantom{A}$formal geometry$\phantom{A}$ | $\phantom{A}$FormalCartSp$\phantom{A}$ | $\phantom{A}$FormalSmoothSet$\phantom{A}$ | $\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$ |
$\phantom{A}$supergeometry$\phantom{A}$ | $\phantom{A}$SuperFormalCartSp$\phantom{A}$ | $\phantom{A}$SuperFormalSmoothSet$\phantom{A}$ | $\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$ |
The study of differential geometry goes back to the study of surfaces embedded into Euclidean space $\mathbb{R}^3$ in
Textbooks include
Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry , Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library
Michael Spivak, A comprehensive introduction to differential geometry (5 Volumes)
Michael Spivak, Calculus on Manifolds (1971)
John Lee, Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)
Loring Tu, Differential Geometry – Connections, Curvature, and Characteristic Classes, Springer 2017 (ISBN:978-3-319-55082-4)
M M Postnikov, Lectures on geometry (6 vols.: 1 “Analytic geometry”, 2 “Linear algebra”, 3 “Diff. manifolds”; 4 “Diff. geometry” (covers extensively fibre bundles and connections); 5 “Lie groups”; 6 “Riemannian geometry”)
With emphasis in G-structures:
With emphasis on Cartan geometry:
Lecture notes include
Brian Conrad, Handouts on Differential Geometry (web)
Liviu Nicolaescu, Lectures on the Geometry of Manifolds, 2018 (pdf)
An introduction with an eye towards applications in physics, specifically to gravity and gauge theory is in
A discussion in the context of Frölicher spaces and diffeological spaces is in
Discussion with emphasis on natural bundles is in
See also
Sigmundur Gudmundsson, An Introduction to Riemannian Geometry (pdf)
Wikipedia, differential geometry
See at higher differential geometry.
For derived differential geometry see
Dominic Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry (web)