Differential geometry and differential topology are two related branches of mathematics that study smooth manifolds , usually with additional structures. They find many applications in physics , especially in the general theory of relativity .
These two sections of mathematics are almost inseparable, and often both sections together are called differential geometry . The difference between these sections is the presence or absence of local invariants. In differential topology, such structures on manifolds are considered that for any pair of points one can find identical neighborhoods, while in differential geometry, generally speaking, there may be local invariants (such as curvature ) that can differ at points.
History
Differential geometry arose and developed in close connection with mathematical analysis, which itself largely grew out of geometry problems. Many geometric concepts preceded the corresponding concepts of analysis. So, for example, the concept of tangent preceded the concept of derivative , the concept of area and volume - the concept of integral .
The emergence of differential geometry dates back to the 18th century and is associated with the names of Euler and Monge . The first consolidated work on surface theory was written by Monge ("An Application of Analysis to Geometry", 1795 ). In 1827, Gauss published the work "General Study of Curved Surfaces", in which he laid the foundations of the theory of surfaces in its modern form. Since then, differential geometry has ceased to be only an application of analysis and has taken an independent place in mathematics.
The discovery of non-Euclidean geometry played a huge role in the development of all geometry, including differential geometry . Riemann in his lecture "On the hypotheses underlying the foundations of geometry" ( 1854 ) laid the foundations of Riemannian geometry , the most developed part of modern differential geometry.
The group-theoretical point of view of Klein set forth in his “ Erlangen Program ” ( 1872 ), that is: geometry — the study of invariants of transformation groups, as applied to differential geometry, was developed by Cartan , who constructed the theory of spaces with projective connection and affine connection .
Differential topology is a much younger branch of mathematics; it begins to develop only at the beginning of the 20th century .
Main subsections of differential geometry and topology
- Differential geometry of curves
- Differential surface geometry
- Riemannian geometry
- Symplectic topology
- Surface theory
- Finsler geometry
Literature
- Dubrovin B.A., Novikov S.P., Fomenko A.T. Modern geometry. Methods and applications. - M .: Nauka, 1986 .-- 760 p.
- Mishchenko A.S., Fomenko A.T. The course of differential geometry and topology. - M .: Moscow State University, 1980 .-- 439 p.
- J. Schwartz. Differential geometry and topology. - M .: Mir, 1970 .-- 223 p.
Resources of the physical and mathematical library of the EqWorld site - “The World of Mathematical Equations” :
- Veblen O. , Whitehead J. Foundations of Differential Geometry = The Foundations of Differential Geometry / Per. from English M. G. Freidina. - M .: IL, 1949 .-- 230 p.
- Huseyn-Zade S. M. Lectures on differential geometry . - M .: Moscow State University, 2001 .-- 54 p.
- Egorov D. F. Works on differential geometry . - M .: Nauka, 1970 .-- 380 p.
- Nomizu K. Lie groups and differential geometry = Lie Groups and Differential Geometry / Per. from English Yu. A. Shub-Sizonenko. - M .: IL, 1960 .-- 128 p.
- Pogorelov A.I. Differential geometry . - 6th ed. - M .: Nauka, 1974 .-- 176 p.
- Rashevsky P.K. Course in differential geometry . - 3rd ed. - M.-L .: GITTL, 1950 .-- 428 p.
- Rozendorn E. R. Problems in differential geometry . - M .: Nauka, 1971. - 64 p.
- Sternberg S. Lectures on Differential Geometry = Lectures on Differential Geometry / Per. from English D.V. Alekseevsky. - M .: Mir, 1970 .-- 412 p.
- Troitsky E.V. Differential geometry and topology . - M .: Moscow State University, 2003 .-- 52 p.
- Finikov S.P. Differential geometry. Course of lectures . - M .: Moscow State University, 1961 .-- 158 p.
- Finikov S.P. Projective differential geometry . - M.-L .: ONTI, 1937 .-- 264 p.
Other jobs:
- Skopenkov A. Fundamentals of differential geometry in interesting problems . - M .: ICMMO, 2008.