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© 2015

Moduli Spaces of Riemannian Metrics

  • First book dealing exclusively with this topic which has hitherto only been treated in original research papers

  • Develops relevant background and explains the ideas involved

  • Short, concise text with topics ranging from classical results right up to the most recent developments

  • Suitable for graduate students with an interest in Riemannian geometry

Textbook

Part of the Oberwolfach Seminars book series (OWS, volume 46)

Table of contents

  1. Front Matter
    Pages I-X
  2. Wilderich Tuschmann, David J. Wraith
    Pages 1-6
  3. Wilderich Tuschmann, David J. Wraith
    Pages 7-16
  4. Wilderich Tuschmann, David J. Wraith
    Pages 17-25
  5. Wilderich Tuschmann, David J. Wraith
    Pages 27-36
  6. Wilderich Tuschmann, David J. Wraith
    Pages 37-47
  7. Wilderich Tuschmann, David J. Wraith
    Pages 49-58
  8. Wilderich Tuschmann, David J. Wraith
    Pages 59-69
  9. Wilderich Tuschmann, David J. Wraith
    Pages 71-87
  10. Wilderich Tuschmann, David J. Wraith
    Pages 89-92
  11. Wilderich Tuschmann, David J. Wraith
    Pages 93-98
  12. Wilderich Tuschmann, David J. Wraith
    Pages 99-101
  13. Back Matter
    Pages 103-123

About this book

Introduction

This book studies certain spaces of Riemannian metrics on both compact and non-compact manifolds. These spaces are defined by various sign-based curvature conditions, with special attention paid to positive scalar curvature and non-negative sectional curvature, though we also consider positive Ricci and non-positive sectional curvature. If we form the quotient of such a space of metrics under the action of the diffeomorphism group (or possibly a subgroup) we obtain a moduli space. Understanding the topology of both the original space of metrics and the corresponding moduli space form the central theme of this book. For example, what can be said about the connectedness or the various homotopy groups of such spaces? We explore the major results in the area, but provide sufficient background so that a non-expert with a grounding in Riemannian geometry can access this growing area of research.

Keywords

Riemannian metrics curvature manifolds moduli spaces topology

Authors and affiliations

  1. 1.Institute for Algebra and GeometryKarlsruher Institut für Technologie KITKarlsruheGermany
  2. 2.Department of Mathematics and StatiNational University of IrelandMaynoothIreland

About the authors

Wilderich Tuschmann's general research interests lie in the realms of global differential geometry, Riemannian geometry, geometric topology, and their applications, including, for example, questions concerning the geometry and topology of nonnegative and almost nonnegative curvature, singular metric spaces, collapsing and Gromov-Hausdorff convergence, analysis and geometry on Alexandrov spaces, geometric finiteness theorems, moduli spaces of Riemannian metrics, transformation groups, geometric bordism invariants, information and quantum information geometry. After his habilitation in mathematics at the University of Leipzig in 2000 he worked as a Deutsche Forschungsgemeinschaft Heisenberg Fellow at Westfälische Wilhems-Universität Münster, and from 2005-2010 he held a professorship at Christian-Albrechts-Universität Kiel. In the fall of 2010 he was appointed professor of mathematics at Karlsruhe Institute of Technology (KIT), a position he currently holds. David Wraith's main mathematical interests concern the existence of Riemannian metrics satisfying various kinds of curvature conditions and their topological implications. Most of his work to date has focused on the existence of positive Ricci curvature metrics. He has worked at the National University of Ireland Maynooth since 1997.

Bibliographic information

Reviews

“This book serves as a comprehensive (yet succinct and accessible) guide to the topology of spaces of Riemannian metrics with a given curvature sign condition. … This is one of the most well-studied aspects of moduli spaces of Riemannian metrics but remains a very active area of research, and the reader will find in this book the current state-of-the-art results on the subject.” (Renato G. Bettiol, Mathematical Reviews, October, 2016)

“The interplay between analysis, geometry, and topology is clearly laid out in this book; analytic invariants are constructed to elucidate the structure of geometric moduli spaces. The book is an elegant and concise introduction to the field that puts a number of discrete papers into a coherent focus. … A useful bibliography of the subject appears at the end.” (Peter B. Gilkey, zbMATH 1336.53002, 2016)