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.
In 1854, B Riemann introduced the notion of curvature for spaces with a family of inner products. There was no significant progress in the general case until 1918, when P Finsler studied the variation problem in regular metric spaces. Around 1926, L Berwald extended Riemann''s notion of curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler''s category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern metric geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov''s Hausdorff convergence theory. Contents: Finsler Spaces; Finsler m Spaces; Co-Area Formula; Isoperimetric Inequalities; Geodesics and Connection; Riemann Curvature; Non-Riemannian Curvatures; Structure Equations; Finsler Spaces of Constant Curvature; Second Variation Formula; Geodesics and Exponential Map; Conjugate Radius and Injectivity Radius; Basic Comparison Theorems; Geometry of Hypersurfaces; Geometry of Metric Spheres; Volume Comparison Theorems; Morse Theory of Loop Spaces; Vanishing Theorems for Homotopy Groups; Spaces of Finsler Spaces. Readership: Graduate students and researchers in geometry and physics.
In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces. In 1854, B. Riemann introduced the Riemann curvature for Riemannian spaces in his ground-breaking Habilitationsvortrag. Thereafter the geometry of these special regular metric spaces is named after him. Riemann also mentioned general regular metric spaces, but he thought that there were nothing new in the general case. In fact, it is technically much more difficult to deal with general regular metric spaces. For more than half century, there had been no essential progress in this direction until P. Finsler did his pioneering work in 1918. Finsler studied the variational problems of curves and surfaces in general regular metric spaces. Some difficult problems were solved by him. Since then, such regular metric spaces are called Finsler spaces. Finsler, however, did not go any further to introduce curvatures for regular metric spaces. He switched his research direction to set theory shortly after his graduation.
This volume features proceedings from the 1995 Joint Summer Research Conference on Finsler Geometry, chaired by S. S. Chern and co-chaired by D. Bao and Z. Shen. The editors of this volume have provided comprehensive and informative "capsules" of presentations and technical reports. This was facilitated by classifying the papers into the following 6 separate sections - 3 of which are applied and 3 are pure: * Finsler Geometry over the reals * Complex Finsler geometry * Generalized Finsler metrics * Applications to biology, engineering, and physics * Applications to control theory * Applications to relativistic field theory Each section contains a preface that provides a coherent overview of the topic and includes an outline of the current directions of research and new perspectives. A short list of open problems concludes each contributed paper. A number of photos are featured in the volumes, for example, that of Finsler. In addition, conference participants are also highlighted.
This comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds.In Part I, the authors discuss differential manifolds, Finsler metrics, the Chern connection, Riemannian and non-Riemannian quantities. Part II is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metrics, conformal transformations, amongst other related topics. The authors made great efforts to ensure that the contents are accessible to senior undergraduate students, graduate students, mathematicians and scientists.
Riemann-Finsler geometry is a subject that concerns manifolds with Finsler metrics, including Riemannian metrics. It has applications in many fields of the natural sciences. Curvature is the central concept in Riemann-Finsler geometry. This invaluable textbook presents detailed discussions on important curvatures such as the Cartan torsion, the S-curvature, the Landsberg curvature and the Riemann curvature. It also deals with Finsler metrics with special curvature or geodesic properties, such as projectively flat Finsler metrics, Berwald metrics, Finsler metrics of scalar flag curvature or isotropic S-curvature, etc. Instructive examples are given in abundance, for further description of some important geometric concepts. The text includes the most recent results, although many of the problems discussed are classical.
This comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds.In Part I, the authors discuss differential manifolds, Finsler metrics, the Chern connection, Riemannian and non-Riemannian quantities. Part II is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metrics, conformal transformations, amongst other related topics. The authors made great efforts to ensure that the contents are accessible to senior undergraduate students, graduate students, mathematicians and scientists.
This volume features proceedings from the 1995 Joint Summer Research Conference on Finsler Geometry, chaired by S. S. Chern and co-chaired by D. Bao and Z. Shen. The editors of this volume have provided comprehensive and informative "capsules" of presentations and technical reports. This was facilitated by classifying the papers into the following 6 separate sections - 3 of which are applied and 3 are pure: * Finsler Geometry over the reals * Complex Finsler geometry * Generalized Finsler metrics * Applications to biology, engineering, and physics * Applications to control theory * Applications to relativistic field theory Each section contains a preface that provides a coherent overview of the topic and includes an outline of the current directions of research and new perspectives. A short list of open problems concludes each contributed paper. A number of photos are featured in the volumes, for example, that of Finsler. In addition, conference participants are also highlighted.
This book gives a contemporary overview of the technologies of single-frequency fiber lasers. The development of single-frequency fiber lasers is one of the most significant achievements in the field of laser photonics over the past two decades. Owing to the crucial demands of a laser sources with highly stable single-frequency operation, narrow linewidth, low noise, scalable to high output power, compact and robustness structure, fiber lasers have been intensively studied since its introduction to the single-frequency laser community and they still continuously proceed to trigger the emergence of new technologies and applications. This book systematically demonstrates the single-frequency fiber laser technologies from fundamental principles to state-of-the-art progress. Details of selected typical applications of single-frequency fiber lasers are also given and discussed. The reader will acquire a good knowledge of the current situation within this important field.
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.
In 1854, B Riemann introduced the notion of curvature for spaces with a family of inner products. There was no significant progress in the general case until 1918, when P Finsler studied the variation problem in regular metric spaces. Around 1926, L Berwald extended Riemann''s notion of curvature to regular metric spaces and introduced an important non-Riemannian curvature using his connection for regular metrics. Since then, Finsler geometry has developed steadily. In his Paris address in 1900, D Hilbert formulated 23 problems, the 4th and 23rd problems being in Finsler''s category. Finsler geometry has broader applications in many areas of science and will continue to develop through the efforts of many geometers around the world. Usually, the methods employed in Finsler geometry involve very complicated tensor computations. Sometimes this discourages beginners. Viewing Finsler spaces as regular metric spaces, the author discusses the problems from the modern metric geometry point of view. The book begins with the basics on Finsler spaces, including the notions of geodesics and curvatures, then deals with basic comparison theorems on metrics and measures and their applications to the Levy concentration theory of regular metric measure spaces and Gromov''s Hausdorff convergence theory. Contents: Finsler Spaces; Finsler m Spaces; Co-Area Formula; Isoperimetric Inequalities; Geodesics and Connection; Riemann Curvature; Non-Riemannian Curvatures; Structure Equations; Finsler Spaces of Constant Curvature; Second Variation Formula; Geodesics and Exponential Map; Conjugate Radius and Injectivity Radius; Basic Comparison Theorems; Geometry of Hypersurfaces; Geometry of Metric Spheres; Volume Comparison Theorems; Morse Theory of Loop Spaces; Vanishing Theorems for Homotopy Groups; Spaces of Finsler Spaces. Readership: Graduate students and researchers in geometry and physics.
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