nLab differential geometry

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 $n$POV

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 $Sh_{(\infty,1)}(ThCartSp)$.

Related concepts

• differential geometry of curves and surfaces

• synthetic differential geometry

• higher differential geometry, derived differential geometry

  • higher differential geometry applied to plain differential geometry

• differential cohesive homotopy type theory

• prequantum geometry, higher prequantum geometry

• D-geometry

• arithmetic differential geometry

• Lie theory

• fiber bundles in physics

• some modern subfields of differential geometry include:

  • symplectic geometry,

  • contact geometry,

  • complex geometry,

  • Riemannian geometry,

  • Finsler geometry?,

  • symmetric spaces,

  • Fréchet manifolds

$\,$

local model	global geometry
Klein geometry	Cartan geometry
Klein 2-geometry	Cartan 2-geometry
higher Klein geometry	higher Cartan geometry

$\,$

geometries of physics

$\phantom{A}$(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$\phantom{A}$

References

Diff geometry of curves and surfaces

The study of differential geometry goes back to the special case of differential geometry of curves and surfaces:

the study of curves and surfaces embedded into Euclidean space $\mathbb{R}^3$:

• Carl Friedrich Gauss, General Investigations of Curved Surfaces (1827) (Gutenberg)

• Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall (1976) $[$pdf$]$

• Kristopher Tapp, Differential Geometry of Curves and Surfaces, Springer (2016) $[$doi:10.1007/978-3-319-39799-3, pdf$]$

• Anton Petrunin, Sergio Zamora Barrera, What is differential geometry: curves and surfaces $[$arXiv:2012.11814$]$

General but traditional diff geometry

• Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry, Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library

• Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, Analysis, manifolds and physics, North Holland (1982, 2001) $[$ISBN:9780444860170$]$

  (with an eye towards mathematical physics)

• 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, book webpage, 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:

• Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall (1964)

With emphasis on Cartan geometry:

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

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

• Theodore Frankel, The Geometry of Physics - An Introduction

• Chris Isham, Modern Differential Geometry for Physicists

A discussion in the context of Frölicher spaces and diffeological spaces is in

• Andreas Kriegl, Peter Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. 618 pages

Discussion with emphasis on natural bundles is in

• Ivan Kolář, Peter Michor, Jan Slovák, Natural operations in differential geometry (pdf)

See also

• Sigmundur Gudmundsson, An Introduction to Riemannian Geometry (pdf)

• Wikipedia, differential geometry

Higher diff geometry

See at higher differential geometry.

Derived diff geometry

For derived differential geometry see

• Dominic Joyce, D-manifolds and d-orbifolds: a theory of derived differential geometry (web)

• Urs Schreiber, Seminar on derived differential geometry