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Lectures on Differential Geometry
EMS Series of Lectures in Mathematics

Iskander A. Taimanov (Sobolev Institute of Mathematics, Novosibirsk, Russia)

Lectures on Differential Geometry

ISBN print 978-3-03719-050-0, ISBN online 978-3-03719-550-5
DOI 10.4171/050
April 2008, 219 pages, softcover, 17.0 x 24.0 cm.
34.00 Euro

Differential geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Modern differential geometry in its turn strongly contributed to modern physics.

This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences.

The text is divided into three parts. The first part covers the basics of curves and surfaces, while the second part is designed as an introduction to smooth manifolds and Riemannian geometry. In particular, Chapter 5 contains short introductions to hyperbolic geometry and geometrical principles of special relativity theory. Here, only a basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and introduces into matrix Lie groups and Lie algebras, representation theory of groups, symplectic and Poisson geometry, and applications of complex analysis in surface theory.

The book is based on lectures the author held repeatedly at Novosibirsk State University. It is addressed to students as well as to anyone who wants to learn the basics of differential geometry.

Keywords: Differential geometry, Riemannian geometry, smooth manifolds, minimal surfaces, Lie groups, representation theory, symplectic and Poisson geometry, finite-dimensional integrable systems

Further Information

Review in Zbl 1143.53003

Review in MR 2406360

MAA Reviews