Riemannian Geometry, third edition
This book is based on a graduate course on Riemannian geometry and analysis on manifolds, held in Paris. It covers the following topics: differential manifolds and elements of global analysis, Riemannian geometry, affine and Riemannian connections, properties of geodesics (including global properties) in general and on special manifolds, the Riemannian curvature tensor, the arc-length and energy, Jacobi vector fields, Riemannian submersions; curvature and its connection with topology, volume and fundamental group; hyperbolic and conformal geometry, manifolds with boundary and Stokes theorem, inequalities by Bishop and Heintze-Karcher, differential forms and cohomology, the Hodge-de Rham theorem, basic spectral geometry and examples, the minimax principle, eigenvalues estimates, Paul Levy’s isoparametric inequality, Riemannian submanifolds, curvature and convexity, minimal surfaces. Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains numerous nontrivial exercises with full solutions at the end. The properties are always illustrated by many detailed examples. The first edition appeared in 1987. For the third edition, some topics on geodesic flow and Lorentzian geometry have been added and worked out in the same spirit.