Download or read book Lectures on Spaces of Nonpositive Curvature written by Werner Ballmann and published by Birkhäuser. This book was released on 2012-12-06 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: Singular spaces with upper curvature bounds and, in particular, spaces of nonpositive curvature, have been of interest in many fields, including geometric (and combinatorial) group theory, topology, dynamical systems and probability theory. In the first two chapters of the book, a concise introduction into these spaces is given, culminating in the Hadamard-Cartan theorem and the discussion of the ideal boundary at infinity for simply connected complete spaces of nonpositive curvature. In the third chapter, qualitative properties of the geodesic flow on geodesically complete spaces of nonpositive curvature are discussed, as are random walks on groups of isometries of nonpositively curved spaces. The main class of spaces considered should be precisely complementary to symmetric spaces of higher rank and Euclidean buildings of dimension at least two (Rank Rigidity conjecture). In the smooth case, this is known and is the content of the Rank Rigidity theorem. An updated version of the proof of the latter theorem (in the smooth case) is presented in Chapter IV of the book. This chapter contains also a short introduction into the geometry of the unit tangent bundle of a Riemannian manifold and the basic facts about the geodesic flow. In an appendix by Misha Brin, a self-contained and short proof of the ergodicity of the geodesic flow of a compact Riemannian manifold of negative curvature is given. The proof is elementary and should be accessible to the non-specialist. Some of the essential features and problems of the ergodic theory of smooth dynamical systems are discussed, and the appendix can serve as an introduction into this theory.
Download or read book Einstein Manifolds written by Arthur L. Besse and published by Springer Science & Business Media. This book was released on 2007-12-03 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.
Download or read book Comparison Geometry written by Karsten Grove and published by Cambridge University Press. This book was released on 1997-05-13 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an up to date work on a branch of Riemannian geometry called Comparison Geometry.
Download or read book An Introduction to Extremal Kahler Metrics written by Gábor Székelyhidi and published by American Mathematical Soc.. This book was released on 2014-06-19 with total page 210 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Geometric Inverse Problems written by Gabriel P. Paternain and published by Cambridge University Press. This book was released on 2023-01-05 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This up-to-date treatment of recent developments in geometric inverse problems introduces graduate students and researchers to an exciting area of research. With an emphasis on the two-dimensional case, topics covered include geodesic X-ray transforms, boundary rigidity, tensor tomography, attenuated X-ray transforms and the Calderón problem. The presentation is self-contained and begins with the Radon transform and radial sound speeds as motivating examples. The required geometric background is developed in detail in the context of simple manifolds with boundary. An in-depth analysis of various geodesic X-ray transforms is carried out together with related uniqueness, stability, reconstruction and range characterization results. Highlights include a proof of boundary rigidity for simple surfaces as well as scattering rigidity for connections. The concluding chapter discusses current open problems and related topics. The numerous exercises and examples make this book an excellent self-study resource or text for a one-semester course or seminar.
Download or read book Einstein Manifolds written by Arthur L. Besse and published by Springer. This book was released on 2007-11-12 with total page 523 pages. Available in PDF, EPUB and Kindle. Book excerpt: Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.
Download or read book Proceedings of the 1981 Shanghai Symposium on Differential Geometry and Differential Equations written by and published by . This book was released on 1984 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Proceedings of the International Conference on Complex Geometry and Related Fields written by Zhijie Chen and published by American Mathematical Soc.. This book was released on 2007 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: In commemoration and celebration of the tenth anniversary of the Institute of Mathematics at East China Normal University, an International Conference on complex geometry and related fields recently convened. This collection presents some of the conference highlights, dealing with various and significant topics of differential and algebraic geometry, while exploring their connections to number theory and mathematical physics. Information for our distributors: Titles in this series are co-published with International Press, Cambridge, MA.
Download or read book Differential Geometry Partial Differential Equations on Manifolds written by Robert Everist Greene and published by American Mathematical Soc.. This book was released on 1993 with total page 585 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 1 begins with a problem list by S.T. Yau, successor to his 1980 list ( Sem
Download or read book Contemporary Geometry written by Hung-Hsi Wu and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 483 pages. Available in PDF, EPUB and Kindle. Book excerpt: Early one morning in April of 1987, the Chinese mathematician J. -Q. Zhong died unexpectedly of a heart attack in New York. He was then near the end of a one-year visit in the United States. When news of his death reached his Chinese-American friends, it was immediately decided by one and all that something should be done to preserve his memory. The present volume is an outgrowth of this sentiment. His friends in China have also established a Zhong Jia-Qing Memorial Fund, which has since twice awarded the Zhong Jia-Qing prizes for Chinese mathematics graduate students. It is hoped that at least part of the reasons for the esteem and affection in which he was held by all who knew him would come through in the succeeding pages of this volume. The three survey chapters by Li and Treibergs, Lu, and Siu (Chapters 1-3) all center around the areas of mathematics in which Zhong made noteworthy contributions. In addition to putting Zhong's mathematical contributions in perspective, these articles should be useful also to a large segment of the mathematical community; together they give a coherent picture of a sizable portion of contemporary geometry. The survey of Lu differs from the other two in that it gives a firsthand account of the work done in the People's Republic of China in several complex variables in the last four decades.
Download or read book Einstein Equations Physical and Mathematical Aspects of General Relativity written by Sergio Cacciatori and published by Springer Nature. This book was released on 2019-11-23 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is based on lectures given at the first edition of the Domoschool, the International Alpine School in Mathematics and Physics, held in Domodossola, Italy, in July 2018. It is divided into two parts. Part I consists of four sets of lecture notes. These are extended versions of lectures given at the Domoschool, written by well-known experts in mathematics and physics related to General Relativity. Part II collects talks by selected participants, focusing on research related to General Relativity.
Download or read book Elliptic and Parabolic Methods in Geometry written by Ben Chow and published by CRC Press. This book was released on 1996-10-15 with total page 212 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book documents the results of a workshop held at the Geometry Center (University of Minnesota, Minneapolis) and captures the excitement of the week.
Download or read book Degeneration of Riemannian metrics under Ricci curvature bounds written by Jeff Cheeger and published by Edizioni della Normale. This book was released on 2001-10-01 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: These notes are based on the Fermi Lectures delivered at the Scuola Normale Superiore, Pisa, in June 2001. The principal aim of the lectures was to present the structure theory developed by Toby Colding and myself, for metric spaces which are Gromov-Hausdorff limits of sequences of Riemannian manifolds which satisfy a uniform lower bound of Ricci curvature. The emphasis in the lectures was on the “non-collapsing” situation. A particularly interesting case is that in which the manifolds in question are Einstein (or Kähler-Einstein). Thus, the theory provides information on the manner in which Einstein metrics can degenerate.
Download or read book Comparison Theorems in Riemannian Geometry written by Jeff Cheeger and published by Newnes. This book was released on 2009-01-15 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: Comparison Theorems in Riemannian Geometry
Download or read book Riemannian Geometry During the Second Half of the Twentieth Century written by Marcel Berger and published by American Mathematical Soc.. This book was released on 2000 with total page 206 pages. Available in PDF, EPUB and Kindle. Book excerpt: During its first hundred years, Riemannian geometry enjoyed steady, but undistinguished growth as a field of mathematics. In the last fifty years of the twentieth century, however, it has exploded with activity. Berger marks the start of this period with Rauch's pioneering paper of 1951, which contains the first real pinching theorem and an amazing leap in the depth of the connection between geometry and topology. Since then, the field has become so rich that it is almost impossible for the uninitiated to find their way through it. Textbooks on the subject invariably must choose a particular approach, thus narrowing the path. In this book, Berger provides a remarkable survey of the main developments in Riemannian geometry in the second half of the last fifty years. One of the most powerful features of Riemannian manifolds is that they have invariants of (at least) three different kinds. There are the geometric invariants: topology, the metric, various notions of curvature, and relationships among these. There are analytic invariants: eigenvalues of the Laplacian, wave equations, Schrödinger equations. There are the invariants that come from Hamiltonian mechanics: geodesic flow, ergodic properties, periodic geodesics. Finally, there are important results relating different types of invariants. To keep the size of this survey manageable, Berger focuses on five areas of Riemannian geometry: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, especially Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the study of periodic geodesics and the geodesic flow. Other topics are treated in less detail in a separate section. While Berger's survey is not intended for the complete beginner (one should already be familiar with notions of curvature and geodesics), he provides a detailed map to the major developments of Riemannian geometry from 1950 to 1999. Important threads are highlighted, with brief descriptions of the results that make up that thread. This supremely scholarly account is remarkable for its careful citations and voluminous bibliography. If you wish to learn about the results that have defined Riemannian geometry in the last half century, start with this book.
Download or read book Mathematical Reviews written by and published by . This book was released on 2003 with total page 1596 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Perspectives In Scalar Curvature In 2 Volumes written by Mikhail L Gromov and published by World Scientific. This book was released on 2022-12-19 with total page 1635 pages. Available in PDF, EPUB and Kindle. Book excerpt: Volume I contains a long article by Misha Gromov based on his many years of involvement in this subject. It came from lectures delivered in Spring 2019 at IHES. There is some background given. Many topics in the field are presented, and many open problems are discussed. One intriguing point here is the crucial role played by two seemingly unrelated analytic means: index theory of Dirac operators and geometric measure theory.Very recently there have been some real breakthroughs in the field. Volume I has several survey articles written by people who were responsible for these results.For Volume II, many people in areas of mathematics and physics, whose work is somehow related to scalar curvature, were asked to write about this in any way they pleased. This gives rise to a wonderful collection of articles, some with very broad and historical views, others which discussed specific fascinating subjects.These two books give a rich and powerful view of one of geometry's very appealing sides.
