# nLab differential geometry of curves and surfaces

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

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

infinitesimal cohesion

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)

# Contents

## Idea

While modern differential geometry has become a very general subject, studying smooth manifolds in full generality, the origins of the subject (before general relativity came along) lie in the study of just curves and surfaces embedded in a Cartesian space/Euclidean space.

Expositions of this differential geometry of curves and surfaces tend to retain some of the historical ideosyncracies (such as the notion of jerk) which are rarely seen in more general texts on differential geometry.

(…)

Historical origins:

Modern accounts:

• 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$]$