# Lectures on Differential Geometry

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*Iskander A. Taimanov*

A publication of the European Mathematical Society

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 the
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 regularly at Novosibirsk
State University. It is addressed to students as well as anyone who wants to
learn the basics of differential geometry.

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

#### Readership

Graduate students and research mathematicians interested in differential geometry

#### Reviews & Endorsements

...this is a wonderfully diverse and challenging introduction to one of the central topics in both mathematics and physics. Its certainly a terrific text to keep handy as a supplement when teaching the subject. And of course, it would a very handy text for graduate students come prelim time.

-- MAA Reviews