Differential Geometry

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.

這本書結合了傳統和現代的微分幾何方法。在介紹流形的黎曼幾何之前，詳細討論了曲線和曲面的性質。

關於平面曲線的微分幾何章節考慮了曲線的局部和全局性質、進化曲線和包絡線、以及仿射和射影微分幾何。討論了曲面的各種高斯曲率方法。首先詳細討論了二維曲面的曲率張量、共軛點和拉普拉斯-貝爾特拉米算子，這有助於在多維情況下研究它們。另外還有一章專門介紹了李群的微分幾何。

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).

維克多·普拉索洛夫（Victor Prasolov），出生於1956年，是莫斯科獨立大學的數學永久教師。他與Springer出版社合作出版了兩本書，分別是《多項式》和《代數曲線：走向模空間》（與M. E. Kazaryan和S. K. Lando合著），以及與AMS合作出版了八本書，包括《線性代數的問題與定理》、《直觀拓撲學》、《結、鏈、辮子和三維流形》（與A. B. Sossinsky合著）和《橢圓函數和橢圓積分》（與Yu. Solovyev合著）。