Homogeneous Finsler Spaces

ISBN-13: 9781461442431 / Angielski / Twarda / 2012 / 242 str.

Shaoqiang Deng

Opis

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.​

Recenzje

From the reviews:

"The main purpose of this book is to show how ideas from Lie theory have spread to Finsler geometry. This book is the first one in the field of homogeneous Finsler spaces. ... Finsler geometry has been developing rapidly, but this book may give a new spirit to Finsler geometry from the view of Lie theory, and it can be highly recommended to anyone who wants to study Finsler geometry from this point of view." (Hamid Reza Salimi Moghaddam, Mathematical Reviews, June, 2013)

"The aim of the present book is to introduce the aspects of Finsler geometry that can be expressed in terms of Lie theory, having as permanent example the case of homogeneous/symmetric Riemannian manifolds. In this way, new very interesting facts are produced by non-Riemannian tools and geometrical objects like flag and S-curvature. ... this book will be of great interest for a large number of geometers." (Radu Miron, Zentralblatt MATH, Vol. 1253, 2013)

Spis treści

Preface.- Acknowledgements.- 1. Introduction to Finsler Geometry.- 2. Lie Groups and Homogenous Spaces.- 3. The Group of Isometries.- 4. Homogeneous Finsler Spaces.- 5. Symmetric Finsler Spaces.- 6. Weakly Symmetric Finsler Spaces.- 7. Homogeneous Randers Spaces.- References.- Index.​  

Blurb

This book is a unique addition to the existing literature in the field of Finsler geometry. This is the first monograph to deal exclusively with homogeneous Finsler geometry and to make serious use of Lie theory in the study of this rapidly developing field. The increasing activity in Finsler geometry can be attested in large part to the driving influence of S.S. Chern, its proven use in many fields of scientific study such as relativity, optics, geosciences, mathematical biology, and psychology, and its promising reach to real-world applications.  This work has potential for broad readership; it is a valuable resource not only for specialists of Finsler geometry, but also for differential geometers who are familiar with Lie theory, transformation groups, and homogeneous spaces. The exposition is rigorous, yet gently engages the reader—student and researcher alike—in developing a ground level understanding of the subject. A one-term graduate course in differential geometry and elementary topology are prerequisites.

In order to enhance understanding, the author gives a detailed introduction and motivation for the topics of each chapter, as well as historical aspects of the subject, numerous well-selected examples, and thoroughly proved main results. Comments for potential further development are presented in Chapters 3–7.   A basic introduction to Finsler geometry is included in Chapter 1;  the essentials of the related classical theory of Lie groups, homogeneous spaces and groups of isometries are presented in Chapters 2–3. Then the author develops the theory of homogeneous spaces within the Finslerian framework. Chapters 4–6 deal with homogeneous, symmetric and weakly symmetric  Finsler spaces. Chapter 7  is entirely devoted to homogeneous Randers spaces,  which are good candidates for real world applications and beautiful illustrators of the developed theory.