Moduli Spaces of Riemannian Metrics

1. Front Matter

   Pages I-X

   PDF

2. Spaces of metrics

   • Wilderich Tuschmann, David J. Wraith

   Pages 1-6

3. Clifford algebras and spin

   • Wilderich Tuschmann, David J. Wraith

   Pages 7-16

4. Dirac operators and index theorems

   • Wilderich Tuschmann, David J. Wraith

   Pages 17-25

5. Early results about the space of positive scalar curvature metrics

   • Wilderich Tuschmann, David J. Wraith

   Pages 27-36

6. The Kreck-Stolz s-invariant

   • Wilderich Tuschmann, David J. Wraith

   Pages 37-47

7. Applications of the s-invariant

   • Wilderich Tuschmann, David J. Wraith

   Pages 49-58

8. The Observer Moduli Space

   • Wilderich Tuschmann, David J. Wraith

   Pages 59-69

9. A survey of other results

   • Wilderich Tuschmann, David J. Wraith

   Pages 71-87

10. Moduli spaces of Riemannian metrics with negative sectional curvature

    • Wilderich Tuschmann, David J. Wraith

    Pages 89-92

11. Non-negative sectional curvature moduli spaces on open manifolds

    • Wilderich Tuschmann, David J. Wraith

    Pages 93-98

12. The Klingenberg-Sakai conjecture and the space of positively pinched metrics

    • Wilderich Tuschmann, David J. Wraith

    Pages 99-101

13. Back Matter

    Pages 103-123

    PDF

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.

• Riemannian metrics
• curvature
• manifolds
• moduli spaces
• topology

“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)

• Institute for Algebra and Geometry, Karlsruher Institut für Technologie KIT, Karlsruhe, Germany

  Wilderich Tuschmann

• Department of Mathematics and Stati, National University of Ireland, Maynooth, Ireland

  David J. Wraith

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.

Policies and ethics