Riemannian geometry : a modern introduction / Isaac Chavel

I. Riemannian manifolds, II. Riemannian curvature, III. Riemannian volume, IV. Riemannian coverings, V. Surfaces, VI. Isoperimetric inequalities (constant curvature), VII. The kinematic density, VIII. Isoperimetric inequalities (variable curvature), IX. Comparison and finiteness theorems.

Bibliogr. p. 449-464. Index

Chapters 1 to 5 present basic notions of classical Riemannian geometry, such as connections, curvature, geodesics, metrics, Jacobi fields, submanifolds, fundamental group, coverings, Gauss-Bonnet theorem of surfaces etc. The sixth chapter treats the Brunn-Minkowski theorem, the solvability of the Neumann problem in ℝ n , Gromov’s uniqueness proof.

The seventh chapter is devoted to differential geometry of analytical dynamics, Berger-Kazdan inequalities, Santalo’s formula. The eighth chapter treats Goke’s isoperimetric inequality, Buser’s isoperimetric inequality, and isoperimetric constants. The ninth chapter is devoted to comparision and finiteness theorems: Rauch’s comparison theorem, triangle comparison theorem, Cheeger’s finiteness theorem etc. There are sections of notes and exercises at the end of each chapter of the book. The bibliography contains all classical works on this subject. (Zentralblatt)