differential geometry

For a radically modern point of view see parts of the nlab entry differential 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.

Modern subfields of differential geometry include symplectic geometry, contact geometry, Riemannian geometry, Finsler, pseudoriemannian, symmetric spaces, Fréchet manifolds. There are alterantive foundations possible, for example the synthetic differential geometry.

- Bernard F. Schutz,
*Geometrical methods of mathematical physics*(elementary intro) amazon, google - S. Helgason,
*Differential geometry, Lie groups and symmetric spaces* - Shoshichi Kobayashi, Katsumi Nomizu,
*Foundations of differential geometry*, Wiley Classics Library - 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”) - R. Sharpe,
*Differential geometry – Cartan’s generalization of Klein’s Erlagen program*, Springer (1997) - Shlomo Sternberg,
*Lectures on differential geometry* - Andreas Kriegl, Peter W. Michor,
*A convenient setting of global analysis*, Math. Surveys and Monographs**53**, Amer. Math. Soc. 1997. 618 pages, pdf - Ivan Kolár, Jan Slovák, Peter W. Michor,
*Natural operations in differential geometry*, Springer 1993, vi+434 pp. pdf, review - Peter Olver,
*Equivalence, invariants, and symmetry*, Cambridge Univ. Press 1995;*Applications of Lie groups to differential equations*, Springer. - V. I. Arnold,
*Mathematical methods of classical mechanics*, Springer (1989). - wikipedia en, ru
- A. Скопенков, Основы дифференциальной геометрии в интересных задачах. М.: МЦНМО, 2008 (arxiv pdf)
- Yvonne Choquet-Bruhat, Cecile Dewitt-Morette,
*Analysis, manifolds and physics*, 1982 and 2001; part II:*92 applications*, 1989

Last revised on September 15, 2011 at 00:55:58. See the history of this page for a list of all contributions to it.