Author: Gary Richard JensenPublisher:
ISBN:
Category :
Languages : en
Pages : 146
Book Description
Homogeneous Einstein Spaces
Author: Gary Richard JensenAn Introduction to Lie Groups and the Geometry of Homogeneous Spaces
Author: Andreas ArvanitogeōrgosPublisher: American Mathematical Soc.
ISBN: 0821827782
Category : Homogeneous spaces
Languages : en
Pages : 162
Book Description
It is remarkable that so much about Lie groups could be packed into this small book. But after reading it, students will be well-prepared to continue with more advanced, graduate-level topics in differential geometry or the theory of Lie groups. The theory of Lie groups involves many areas of mathematics. In this book, Arvanitoyeorgos outlines enough of the prerequisites to get the reader started. He then chooses a path through this rich and diverse theory that aims for an understanding of the geometry of Lie groups and homogeneous spaces. In this way, he avoids the extra detail needed for a thorough discussion of other topics. Lie groups and homogeneous spaces are especially useful to study in geometry, as they provide excellent examples where quantities (such as curvature) are easier to compute. A good understanding of them provides lasting intuition, especially in differential geometry. The book is suitable for advanced undergraduates, graduate students, and research mathematicians interested in differential geometry and neighboring fields, such as topology, harmonic analysis, and mathematical physics.
Einstein Manifolds
Author: Arthur L. BessePublisher: Springer Science & Business Media
ISBN: 3540741208
Category : Mathematics
Languages : en
Pages : 529
Book Description
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.
Projective Duality and Homogeneous Spaces
Author: Evgueni A. TevelevPublisher: Springer Science & Business Media
ISBN: 3540269576
Category : Mathematics
Languages : en
Pages : 257
Book Description
Projective duality is a very classical notion naturally arising in various areas of mathematics, such as algebraic and differential geometry, combinatorics, topology, analytical mechanics, and invariant theory, and the results in this field were until now scattered across the literature. Thus the appearance of a book specifically devoted to projective duality is a long-awaited and welcome event. Projective Duality and Homogeneous Spaces covers a vast and diverse range of topics in the field of dual varieties, ranging from differential geometry to Mori theory and from topology to the theory of algebras. It gives a very readable and thorough account and the presentation of the material is clear and convincing. For the most part of the book the only prerequisites are basic algebra and algebraic geometry. This book will be of great interest to graduate and postgraduate students as well as professional mathematicians working in algebra, geometry and analysis.
Homogeneous Finsler Spaces
Author: Shaoqiang DengPublisher: Springer Science & Business Media
ISBN: 1461442443
Category : Mathematics
Languages : en
Pages : 250
Book Description
Homogeneous Finsler Spaces is the first book to emphasize the relationship between Lie groups and Finsler geometry, and the first to show the validity in using Lie theory for the study of Finsler geometry problems. This book contains a series of new results obtained by the author and collaborators during the last decade. The topic of Finsler geometry has developed rapidly in recent years. One of the main reasons for its surge in development is its use in many scientific fields, such as general relativity, mathematical biology, and phycology (study of algae). This monograph introduces the most recent developments in the study of Lie groups and homogeneous Finsler spaces, leading the reader to directions for further development. The book contains many interesting results such as a Finslerian version of the Myers-Steenrod Theorem, the existence theorem for invariant non-Riemannian Finsler metrics on coset spaces, the Berwaldian characterization of globally symmetric Finsler spaces, the construction of examples of reversible non-Berwaldian Finsler spaces with vanishing S-curvature, and a classification of homogeneous Randers spaces with isotropic S-curvature and positive flag curvature. Readers with some background in Lie theory or differential geometry can quickly begin studying problems concerning Lie groups and Finsler geometry.
Einstein's Apple: Homogeneous Einstein Fields
Author: Engelbert L SchuckingPublisher: World Scientific
ISBN: 9814630098
Category : Science
Languages : en
Pages : 314
Book Description
We lift a veil of obscurity from a branch of mathematical physics in a straightforward manner that can be understood by motivated and prepared undergraduate students as well as graduate students specializing in relativity. Our book on 'Einstein Fields' clarifies Einstein's very first principle of equivalence (1907) that is the basis of his theory of gravitation. This requires the exploration of homogeneous Riemannian manifolds, a program that was suggested by Elie Cartan in 'Riemannian Geometry in an Orthogonal Frame,' a 2001 World Scientific publication.Einstein's first principle of equivalence, the key to his General Relativity, interprets homogeneous fields of acceleration as gravitational fields. The general theory of these 'Einstein Fields' is given for the first time in our monograph and has never been treated in such exhaustive detail. This study has yielded significant new insights to Einstein's theory. The volume is heavily illustrated and is accessible to well-prepared undergraduate and graduate students as well as the professional physics community.
Finsler Geometry
Author: Xinyue ChengPublisher: Springer Science & Business Media
ISBN: 3642248888
Category : Mathematics
Languages : en
Pages : 149
Book Description
"Finsler Geometry: An Approach via Randers Spaces" exclusively deals with a special class of Finsler metrics -- Randers metrics, which are defined as the sum of a Riemannian metric and a 1-form. Randers metrics derive from the research on General Relativity Theory and have been applied in many areas of the natural sciences. They can also be naturally deduced as the solution of the Zermelo navigation problem. The book provides readers not only with essential findings on Randers metrics but also the core ideas and methods which are useful in Finsler geometry. It will be of significant interest to researchers and practitioners working in Finsler geometry, even in differential geometry or related natural fields. Xinyue Cheng is a Professor at the School of Mathematics and Statistics of Chongqing University of Technology, China. Zhongmin Shen is a Professor at the Department of Mathematical Sciences of Indiana University Purdue University, USA.
Einstein Spaces
Author: A. Z. PetrovPublisher: Elsevier
ISBN: 1483151840
Category : Science
Languages : en
Pages : 427
Book Description
Einstein Spaces presents the mathematical basis of the theory of gravitation and discusses the various spaces that form the basis of the theory of relativity. This book examines the contemporary development of the theory of relativity, leading to the study of such problems as gravitational radiation, the interaction of fields, and the behavior of elementary particles in a gravitational field. Organized into nine chapters, this book starts with an overview of the principles of the special theory of relativity, with emphasis on the mathematical aspects. This text then discusses the need for a general classification of all potential gravitational fields, and in particular, Einstein spaces. Other chapters consider the gravitational fields in empty space, such as in a region where the energy-momentum tensor is zero. The final chapter deals with the problem of the limiting conditions in integrating the gravitational field equations. Physicists and mathematicians will find this book useful.
Naturally Reductive Metrics and Einstein Metrics on Compact Lie Groups
Author: J. E. D'AtriPublisher: American Mathematical Soc.
ISBN: 0821822152
Category : Mathematics
Languages : en
Pages : 80
Book Description
The first part of this paper constructs a class of naturally reductive metrics on compact Lie groups and shows that all naturally reductive left invariant metrics are of this type if the group is simple. The second part analyzes the question of when these metrics are Einstein and gives many new examples. In doing this, certain facts are established about the ratios of the Killing forms of a Lie algebra and a subalgebra. Finally, some results are obtained for noncompact groups and more general compact homogeneous spaces.
The Large Scale Structure of Space-Time
Author: S. W. HawkingPublisher: Cambridge University Press
ISBN: 1139810952
Category : Science
Languages : en
Pages : 406
Book Description
Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.