This book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces.
The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.
Arvustused
All chapters are supplemented with solutions of the problems scattered throughout the text. Designed as a text for a lecturer course on the subject, it is perfect and can be recommended for students interested in this classical field. (Ivailo. M. Mladenov, zbMATH 1498.53001, 2022)
Curves in the Plane.- Curves in Space.- Surfaces in
Space.- Hypersurfaces in Rn+1.- Connections.- Riemannian Manifolds.- Lie
Groups.- Comparison Theorems.- Curvature and Topology.- Laplacian.-
Appendix.- Bibliography.- Index.
Victor Prasolov, born 1956, is a permanent teacher of mathematics at the Independent University of Moscow. He published two books with Springer, Polynomials and Algebraic Curves. Towards Moduli Spaces (jointly with M. E. Kazaryan and S. K. Lando) and eight books with AMS, including Problems and Theorems in Linear Algebra, Intuitive Topology, Knots, Links, Braids, and 3-Manifolds (jointly with A. B. Sossinsky), and Elliptic Functions and Elliptic Integrals (jointly with Yu. Solovyev).
