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 "...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 Table of Contents Part I. Curves and surfaces  Theory of curves
 Theory of surfaces
Part II. Riemannian geometry  Smooth manifolds
 Riemannian manifolds
 The Lobachevskii plane and the Minkowski space
Part III. Supplement chapters  Minimal surfaces and complex analysis
 Elements of Lie group theory
 Elements of representation theory
 Elements of Poisson and symplectic geometry
