# nLab differential geometry

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Ingredients

Concepts

Constructions

Examples

Theorems

# 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 $n$POV

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:

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:

$\,$

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

With emphasis on G-structures:

With emphasis on natural bundles:

With emphasis on Cartan geometry:

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

Lecture notes:

An introduction with an eye towards applications in physics, specifically to gravity and gauge theory:

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

### Higher diff geometry

See at higher differential geometry.

### Derived diff geometry

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

Last revised on December 8, 2022 at 03:37:46. See the history of this page for a list of all contributions to it.