derived smooth geometry
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.
See also generalized smooth space.
This includes a sequence of concepts of generalized smooth spaces:
an infinitesimal space is a certain object in ;
some modern subfields of differential geometry include:
|local model||global geometry|
|Klein geometry||Cartan geometry|
|Klein 2-geometry||Cartan 2-geometry|
|higher Klein geometry||higher Cartan geometry|
Michael Spivak, A comprehensive introduction to differential geometry (5 Volumes)
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:
With emphasis on Cartan geometry:
Lecture notes include
Discussion with emphasis on natural bundles is in
Sigmundur Gudmundsson, An Introduction to Riemannian Geometry (pdf)
Wikipedia, differential geometry
See at higher differential geometry.
For derived differential geometry see