Download or read book Differential Geometry of Spray and Finsler Spaces written by Zhongmin Shen and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Differential Geometry of Projectively Flat Finsler Spaces written by Padma Senarath and published by . This book was released on 2003 with total page 334 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Differential Geometry written by Padma Senarath and published by . This book was released on 2010 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to present the basic concepts of Finsler geometry including connections, flag curvature, projective changes, Randers spaces and other types of Finsler spaces. In this book, we introduce a Riemannian space of non-zero constant sectional curvature by considering a locally projectively flat Finsler space and compute two standard Riemannian metrics of non-zero constant sectional curvature by choosing two solutions of a system of partial differential equations. This book also presents two examples of locally projectively flat Randers metrics of scalar curvature by using the Riemannian metric to illustrate the fact that some locally projectively flat Randers metrics of scalar curvature do not have isotropic S-curvature. Finally we give necessary and sufficient conditions for Finsler spaces with various types of (alpha, beta)-metric to be locally projectively flat and determine whether the conditions, a Riemannian metric (alpha) is locally projectively flat and a one-form (beta) is closed, can occur at the same time in the locally projectively flat Finsler spaces with various types of (alpha, beta)-metric.
Download or read book Finsler Geometry written by Xinyue Cheng and published by Springer Science & Business Media. This book was released on 2013-01-29 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: "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.
Download or read book Lie Groups Differential Equations and Geometry written by Giovanni Falcone and published by Springer. This book was released on 2017-09-19 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
Download or read book An Introduction to Finsler Geometry written by Xiaohuan Mo and published by World Scientific. This book was released on 2006 with total page 130 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introductory book uses the moving frame as a tool and develops Finsler geometry on the basis of the Chern connection and the projective sphere bundle. It systematically introduces three classes of geometrical invariants on Finsler manifolds and their intrinsic relations, analyzes local and global results from classic and modern Finsler geometry, and gives non-trivial examples of Finsler manifolds satisfying different curvature conditions.
Download or read book Introduction To Modern Finsler Geometry written by Yi-bing Shen and published by World Scientific Publishing Company. This book was released on 2016-02-25 with total page 406 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book A Sampler of Riemann Finsler Geometry written by David Dai-Wai Bao and published by Cambridge University Press. This book was released on 2004-11 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: These expository accounts treat issues related to volume, geodesics, curvature and mathematical biology, with instructive examples.
Download or read book The Differential Geometry of Finsler Spaces written by Hanno Rund and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: The present monograph is motivated by two distinct aims. Firstly, an endeavour has been made to furnish a reasonably comprehensive account of the theory of Finsler spaces based on the methods of classical differential geometry. Secondly, it is hoped that this monograph may serve also as an introduction to a branch of differential geometry which is closely related to various topics in theoretical physics, notably analytical dynamics and geometrical optics. With this second object in mind, an attempt has been made to describe the basic aspects of the theory in some detail - even at the expense of conciseness - while in the more specialised sections of the later chapters, which might be of interest chiefly to the specialist, a more succinct style has been adopted. The fact that there exist several fundamentally different points of view with regard to Finsler geometry has rendered the task of writing a coherent account a rather difficult one. This remark is relevant not only to the development of the subject on the basis of the tensor calculus, but is applicable in an even wider sense. The extensive work of H. BUSEMANN has opened up new avenues of approach to Finsler geometry which are independent of the methods of classical tensor analysis. In the latter sense, therefore, a full description of this approach does not fall within the scope of this treatise, although its fundamental l significance cannot be doubted.
Download or read book The Geometry of Finsler Spaces an Approach Via Special Finsler Metric written by Sruthy Baby and published by . This book was released on 2019-10-20 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: Finsler geometry is Riemannian geometry without the restriction that the line element be quadratic. It has applications in many field of natural science especially in mechanics, gravitational theory, electromagnetism, information geometry etc. This book presents some work done by the author on the theory of projective change between two Finsler spaces, Conformal change of Douglas space with special Finsler metric, Nonholonomic Frames for Finsler space with special ( α, β) metric, Reversible geodesics of Finslerian space, Complex Finsler space, Rander -conformal change of Finsler spaces, and the curvature properties of Finsler space. The chapters included in this book contains fundamental topic of modern Riemann Finsler geometry, including the notion of curvature, projectively flat metrics, dually flat metrics which are interesting not only for specialists in Finsler Geometry, but for researchers in Riemann Geometry or other field of differential geometry.The book provides readers with essential findings on a special type of Finsler metric, which can be considered as a generalization of Randers metric and square metric.The text includes the most recent topics in Finsler Geometry like Reversible geodesics of Finsler space, R-Complex Finsler space and transformation on Finsler metric.This book shall be of benefit to students in the field of Differential geometry, and will be of interest to physicists and mathematical biologists.
Download or read book Riemann finsler Geometry written by Shiing-shen Chern and published by World Scientific Publishing Company. This book was released on 2005-05-10 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Aspects of Differential Geometry IV written by Esteban Calviño-Louzao and published by Springer Nature. This book was released on 2022-06-01 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: Book IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces {which} are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group R2 is Abelian and the + group\index{ax+b group} is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type surfaces. These are the left-invariant affine geometries on R2. Associating to each Type surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue =-1$ turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type surfaces; these are the left-invariant affine geometries on the + group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere 2. The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.
Download or read book Finsler Geometry Sapporo 2005 written by Makoto Matsumoto and published by . This book was released on 2007 with total page 456 pages. Available in PDF, EPUB and Kindle. Book excerpt: The volume contains surveys and original articles based on the talks given at the 40-th Finsler Symposium on Finsler Geometry held in the period September 9-10, 2005 at Hokkaido Tokai University, Sapporo, Japan. The Symposium's purpose was not only a meeting of the Finsler geometers from Japan and abroad, but also to commemorate the memory of the late Professor Makoto Matsumoto. The papers included in this volume contain fundamental topics of modern Riemann-Finsler geometry, interesting not only for specialists in Finsler geometry, but for researchers in Riemannian geometry or other fields of differential geometry and its applications also.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North America
Download or read book Lectures On Finsler Geometry written by Zhongmin Shen and published by World Scientific. This book was released on 2001-05-22 with total page 323 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Differential Geometry on Complex and Almost Complex Spaces written by Kentarō Yano and published by . This book was released on 1965 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Projective Differential Geometry Old and New written by V. Ovsienko and published by Cambridge University Press. This book was released on 2004-12-13 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. The authors' main goal in this 2005 book is to emphasize connections between classical projective differential geometry and contemporary mathematics and mathematical physics. They also give results and proofs of classic theorems. Exercises play a prominent role: historical and cultural comments set the basic notions in a broader context. The book opens by discussing the Schwarzian derivative and its connection to the Virasoro algebra. One-dimensional projective differential geometry features strongly. Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, and the classical four-vertex theorem. The classical theory of projective hypersurfaces is surveyed and related to some very recent results and conjectures. A final chapter considers various versions of multi-dimensional Schwarzian derivative. In sum, here is a rapid route for graduate students and researchers to the frontiers of current research in this evergreen subject.
Download or read book Differential geometry of special mappings written by Josef Mikeš and published by Univerzita Palackého v Olomouci. This book was released on with total page 677 pages. Available in PDF, EPUB and Kindle. Book excerpt: The collective monograph is devoted to geodesic mappings (i.e. diffeomorphisms preserving geodetics) of Riemannian manifolds and their generalizations. The book also shows related geometric issues and it consists of 18 chapters. The first and the second chapters introduce into differential geometry (of curves and surfaces) and topology. The next four chapters are devoted to the basics of special manifolds with affine connections and their mappings. Chapters 7-12 are an extensive analysis of geodesic mappings (including Einstein and Kähler spaces). Chapters 13-16 are dedicated to rotary, F-planar, holomorphic-projective and almost geodesic mappings. Chapter 17 is a survey of the geometry of Riemann-Finsler spaces. Chapter 18 deals with A-spaces and Klingenberg projective spaces.
