This book is a compendium of survey lectures presented at a conference on Riemannian Geometry sponsored by The Fields Institute for Research in Mathematical Sciences (Waterloo, Canada) in August 1993. Attended by over 80 participants, the aim of the conference was to promote research activity in Riemannian geometry. A select group of internationally established researchers in the field were invited to discuss and present current developments in a selection of contemporary topics in Riemannian geometry. This volume contains four of the five survey lectures presented at the conference.

Features:

• Basic notions of volume and entropy and the difficult and deep relations of these invariants to curvature.
• \(LP\) cohomology, in which the methods combine various areas of mathematics going beyond Riemannian geometry.
• Curvature inequalities from a general point of view, leading to the study of general spaces.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Graduate students and researchers interested in geometry.

Table of Contents

• Lecture Series 1. Gérard Besson, Volumes and entropies
• Preface
• Volumes
• Simplicial volume
• Entropies
• Some results and a new method
• Biblography
• Lecture Series 2. Joachim Lohkamp, Global and local curvatures
• Preface
• Curved balls
• Approximation
• Internal stuctures
• Bibliography
• Lecture Series 3. Pierre Pansu, Introduction to \(L^2\) Betti numbers
• Acknowledgments
• Introduction
• Von-Neumann dimension
• Simplicial \(L^2\) Betti numbers
• Homotopy invariance
• Invariants of discrete groups
• Atiyah's \(L^2\) index theorem
• \(L^\infty\) cohomology and negative curvature
• A vanishing theorem for Kähler hyperbolic manifolds
• Non vanishing theorems for \(L^2\) cohomology
• \(L^2\) index for projectively invariant operators
• Lecture Series 4. Peter Petersen, Comparison geometry problem list
• Bibliography
• Introduction
• Review of techniques in comparison theory
• Results in comparison geometry
• Problems
• Bibliograpy