nLab differential geometry of curves and surfaces

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.

Related concepts

• Euclidean geometry

• curve

  • velocity, areal velocity

  • acceleration

  • jerk

• surface

  • Theorema Egregium?

(…)

References

Historical origins:

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

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$]$

See also:

• Wikipedia, Differential geometry of curves

• Wikipedia, Differential geometry of surfaces