nLab
differential geometry

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Higher geometry

Contents

Scope

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.

Generalized smooth spaces from nnPOV

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 𝒢=\mathcal{G} = CartSp.

This includes a sequence of concepts of generalized smooth spaces:

Similarly, standard models of synthetic differential geometry in higher geometry are modeled on the pre-geometry 𝒢=\mathcal{G} = ThCartSp. To wit, the cohesive topos Sh(ThCartSp)Sh(ThCartSp) is the smooth topos called the Cahiers topos:

local modelglobal geometry
Klein geometryCartan geometry
Klein 2-geometryCartan 2-geometry
higher Klein geometryhigher Cartan geometry

References

Traditional

The study of differential geometry goes back to the study of surfaces embedded into Euclidean space 3\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)

  • 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:

  • R. Sharpe, Differential geometry – Cartan’s generalization of Klein’s Erlagen program, Springer (1997)

Lecture notes include

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

Higher

See at higher differential geometry.

Derived

For derived differential geometry see

Revised on June 26, 2017 05:12:04 by Urs Schreiber (88.77.226.246)