Since the first edition of this graduate textbook in Riemannian Geometry has already been reviewed [S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Springer, Berlin, 1987; MR0909697], this review will focus on those aspects which have changed since the first edition. The second edition of the text added a section on surfaces with constant negative curvature, an introduction to conformal geometry and a proof of the Paul Lévy-Gromov isoperimetric inequality. The third edition adds a new section devoted to pseudo-Riemannian geometry. The geodesic flow from a Hamiltonian point of view is discussed, and an introduction to isosystolic inequalities is presented. This new edition maintains the clear written style of the original, including many illustrations (although fewer than the second edition), examples and exercises (most with solutions).
Reviewer: Borzellino, Joseph E. [form MathSciNet]