synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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.
Theorema Egregium?
(…)
Historical origins:
Modern accounts:
Manfredo P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall (1976) [pdf]
Heinrich W. Guggenheimer, Differential Geometry, Dover (1977) [isbn:9780486634333, ark:/13960/t9t22sk9n]
Victor A. Toponogov, Differential Geometry of Curves and Surfaces – A Concise Guide, Springer (2006) [doi:10.1007/b137116]
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
Last revised on May 18, 2024 at 13:05:06. See the history of this page for a list of all contributions to it.