zbMATH — the first resource for mathematics

Lectures in differential geometry. Transl. from the Russian by Gleb V. Dyatlov. (English) Zbl 1143.53003
EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-050-0/pbk). viii, 211 p. (2008).
This book is an English translation from Russian, based on lecture notes from the author’s geometry course. It is an introduction to modern differential geometry, suitable for advanced undergraduates or beginning graduate students. It is divided into three parts.
The first part discusses the classical theory of curves and surfaces in Euclidean spaces, leading to the Euler-Lagrange equation and the Gauss-Bonnet theorem.
The second part discusses manifold theory and Riemannian geometry, including Riemannian connection, curvature and geodesic. It also studies some hyperbolic and pseudo-Riemannian geometry.
The third part deals with miscellaneous topics in differential geometry, including complex analysis, Lie groups and their representations, Poisson and symplectic geometry. Lie group actions and cotangent bundles lead to a discussion on mechanics and integrable systems.
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53A04 Curves in Euclidean and related spaces
53A05 Surfaces in Euclidean and related spaces
58A05 Differentiable manifolds, foundations
53C20 Global Riemannian geometry, including pinching
22E15 General properties and structure of real Lie groups
53D05 Symplectic manifolds (general theory)
53D17 Poisson manifolds; Poisson groupoids and algebroids
Full Text: DOI