Moduli Spaces of Riemannian Metrics
Moduli Spaces of Riemannian Metrics

Author: Wilderich Tuschmann

Publisher: Springer

Published: 2015-10-14

Total Pages: 127

ISBN-13: 3034809484

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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.

Moduli Spaces of Riemann Surfaces
Moduli Spaces of Riemann Surfaces

Author: Benson Farb

Publisher: American Mathematical Soc.

Published: 2013-08-16

Total Pages: 371

ISBN-13: 0821898876

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Mapping class groups and moduli spaces of Riemann surfaces were the topics of the Graduate Summer School at the 2011 IAS/Park City Mathematics Institute. This book presents the nine different lecture series comprising the summer school, covering a selection of topics of current interest. The introductory courses treat mapping class groups and Teichmüller theory. The more advanced courses cover intersection theory on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable cohomology of mapping class groups, the structure of Torelli groups, and arithmetic mapping class groups. The courses consist of a set of intensive short lectures offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The book should be a valuable resource for graduate students and researchers interested in the topology, geometry and dynamics of moduli spaces of Riemann surfaces and related topics. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Riemannian Geometry
Riemannian Geometry

Author: Gérard Besson

Publisher: American Mathematical Soc.

Published: 1996-01-01

Total Pages: 132

ISBN-13: 9780821871874

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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.

The Geometry of Moduli Spaces of Sheaves
The Geometry of Moduli Spaces of Sheaves

Author: Daniel Huybrechts

Publisher: Cambridge University Press

Published: 2010-05-27

Total Pages: 345

ISBN-13: 1139485822

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This edition has been updated to reflect recent advances in the theory of semistable coherent sheaves and their moduli spaces. The authors review changes in the field and point the reader towards further literature. An ideal text for graduate students or mathematicians with a background in algebraic geometry.

Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures
Riemannian Metrics of Constant Mass and Moduli Spaces of Conformal Structures

Author: Lutz Habermann

Publisher: Springer

Published: 2007-05-06

Total Pages: 123

ISBN-13: 3540444432

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This monograph deals with recent questions of conformal geometry. It provides in detail an approach to studying moduli spaces of conformal structures, using a new canonical metric for conformal structures. This book is accessible to readers with basic knowledge in differential geometry and global analysis. It addresses graduates and researchers.

Gauge Field Theory and Complex Geometry
Gauge Field Theory and Complex Geometry

Author: Yuri I. Manin

Publisher: Springer Science & Business Media

Published: 1997-05-20

Total Pages: 368

ISBN-13: 9783540613787

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From the reviews: "... focused mainly on complex differential geometry and holomorphic bundle theory. This is a powerful book, written by a very distinguished contributor to the field" (Contemporary Physics )"the book provides a large amount of background for current research across a spectrum of field. ... requires effort to read but it is worthwhile and rewarding" (New Zealand Math. Soc. Newsletter) " The contents are highly technical and the pace of the exposition is quite fast. Manin is an outstanding mathematician, and writer as well, perfectly at ease in the most abstract and complex situation. With such a guide the reader will be generously rewarded!" (Physicalia) This new edition includes an Appendix on developments of the last 10 years, by S. Merkulov.

An Introduction to Extremal Kahler Metrics
An Introduction to Extremal Kahler Metrics

Author: Gábor Székelyhidi

Publisher: American Mathematical Soc.

Published: 2014-06-19

Total Pages: 210

ISBN-13: 1470410478

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A basic problem in differential geometry is to find canonical metrics on manifolds. The best known example of this is the classical uniformization theorem for Riemann surfaces. Extremal metrics were introduced by Calabi as an attempt at finding a higher-dimensional generalization of this result, in the setting of Kähler geometry. This book gives an introduction to the study of extremal Kähler metrics and in particular to the conjectural picture relating the existence of extremal metrics on projective manifolds to the stability of the underlying manifold in the sense of algebraic geometry. The book addresses some of the basic ideas on both the analytic and the algebraic sides of this picture. An overview is given of much of the necessary background material, such as basic Kähler geometry, moment maps, and geometric invariant theory. Beyond the basic definitions and properties of extremal metrics, several highlights of the theory are discussed at a level accessible to graduate students: Yau's theorem on the existence of Kähler-Einstein metrics, the Bergman kernel expansion due to Tian, Donaldson's lower bound for the Calabi energy, and Arezzo-Pacard's existence theorem for constant scalar curvature Kähler metrics on blow-ups.

Comparison Geometry
Comparison Geometry

Author: Karsten Grove

Publisher: Cambridge University Press

Published: 1997-05-13

Total Pages: 280

ISBN-13: 9780521592222

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This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.

Recent Developments in Pseudo-Riemannian Geometry
Recent Developments in Pseudo-Riemannian Geometry

Author: Dmitriĭ Vladimirovich Alekseevskiĭ

Publisher: European Mathematical Society

Published: 2008

Total Pages: 556

ISBN-13: 9783037190517

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This book provides an introduction to and survey of recent developments in pseudo-Riemannian geometry, including applications in mathematical physics, by leading experts in the field. Topics covered are: Classification of pseudo-Riemannian symmetric spaces Holonomy groups of Lorentzian and pseudo-Riemannian manifolds Hypersymplectic manifolds Anti-self-dual conformal structures in neutral signature and integrable systems Neutral Kahler surfaces and geometric optics Geometry and dynamics of the Einstein universe Essential conformal structures and conformal transformations in pseudo-Riemannian geometry The causal hierarchy of spacetimes Geodesics in pseudo-Riemannian manifolds Lorentzian symmetric spaces in supergravity Generalized geometries in supergravity Einstein metrics with Killing leaves The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights the interesting new geometric phenomena, which naturally arise in the indefinite metric case. The reader finds a description of the present state of the art in the field as well as open problems, which can stimulate further research.