Taimanov I.A. Lectures on Differential Geometry

European MathematicalSociety, 2008. — 219 p. — (EMS Series of Lectures in Mathematics). — ISBN 978-3-03719-050-0.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 when, for instance, at the beginning of the 20th century it was discovered by Einstein that a gravitational field is just a pseudo-Riemannian metric on space time. The basic equations of gravity theory were written in terms of the curvature of a metric, which is a geometric quantity. More recently the modern theory of elementary particles was based on gauge fields, which mathematically are connections on fiber bundles.
In this book we attempt to give 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.Curves and surfaces
Theory of curves
Theory of surfaces
Riemannian geometry
Smooth manifolds
Riemannian manifolds
The Lobachevskii plane and the Minkowski space
Supplement chapters
Minimal surfaces and complex analysis
Elements of Lie group theory
Elements of representation theory
Elements of Poisson and symplectic geometry