An Introduction To Finsler Geometry

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An Introduction To Riemann Finsler Geometry

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Author : D. Bao
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

An Introduction To Riemann Finsler Geometry written by D. Bao and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.

An Introduction To Riemann Finsler Geometry

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Author : David Dai-Wai Bao
language : en
Publisher: Springer Science & Business Media
Release Date : 2000-03-17

An Introduction To Riemann Finsler Geometry written by David Dai-Wai Bao and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-03-17 with Mathematics categories.

This book focuses on the elementary but essential problems in Riemann-Finsler Geometry, which include a repertoire of rigidity and comparison theorems, and an array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. "This book offers the most modern treatment of the topic ..." EMS Newsletter.

Comparison Finsler Geometry

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Author : Shin-ichi Ohta
language : en
Publisher: Springer Nature
Release Date : 2021-10-09

Comparison Finsler Geometry written by Shin-ichi Ohta and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-10-09 with Mathematics categories.

This monograph presents recent developments in comparison geometry and geometric analysis on Finsler manifolds. Generalizing the weighted Ricci curvature into the Finsler setting, the author systematically derives the fundamental geometric and analytic inequalities in the Finsler context. Relying only upon knowledge of differentiable manifolds, this treatment offers an accessible entry point to Finsler geometry for readers new to the area. Divided into three parts, the book begins by establishing the fundamentals of Finsler geometry, including Jacobi fields and curvature tensors, variation formulas for arc length, and some classical comparison theorems. Part II goes on to introduce the weighted Ricci curvature, nonlinear Laplacian, and nonlinear heat flow on Finsler manifolds. These tools allow the derivation of the Bochner–Weitzenböck formula and the corresponding Bochner inequality, gradient estimates, Bakry–Ledoux’s Gaussian isoperimetric inequality, and functional inequalities in the Finsler setting. Part III comprises advanced topics: a generalization of the classical Cheeger–Gromoll splitting theorem, the curvature-dimension condition, and the needle decomposition. Throughout, geometric descriptions illuminate the intuition behind the results, while exercises provide opportunities for active engagement. Comparison Finsler Geometry offers an ideal gateway to the study of Finsler manifolds for graduate students and researchers. Knowledge of differentiable manifold theory is assumed, along with the fundamentals of functional analysis. Familiarity with Riemannian geometry is not required, though readers with a background in the area will find their insights are readily transferrable.

An Introduction To Finsler Geometry

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Author : Xiaohuan Mo
language : en
Publisher: World Scientific
Release Date : 2006

An Introduction To Finsler Geometry written by Xiaohuan Mo and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006 with Mathematics categories.

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. Book jacket.

The Differential Geometry Of Finsler Spaces

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Author : Hanno Rund
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

The Differential Geometry Of Finsler Spaces written by Hanno Rund and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

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.

Riemann Finsler Geometry

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Author : Shiing-Shen Chern
language : en
Publisher: World Scientific
Release Date : 2005

Riemann Finsler Geometry written by Shiing-Shen Chern and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.

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 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 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. Graduate students and researchers in differential geometry.

Differential Geometry Of Spray And Finsler Spaces

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Author : Zhongmin Shen
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-03-14

Differential Geometry Of Spray And Finsler Spaces written by Zhongmin Shen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-14 with Mathematics categories.

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.

Homogeneous Finsler Spaces

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Author : Shaoqiang Deng
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-08-01

Homogeneous Finsler Spaces written by Shaoqiang Deng and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-08-01 with Mathematics categories.

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.​

Finsler Geometry

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Author : Xinyue Cheng
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-01-29

Finsler Geometry written by Xinyue Cheng and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-29 with Mathematics categories.

"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.

Initiation To Global Finslerian Geometry

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Author : Hassan Akbar-Zadeh
language : en
Publisher: Elsevier
Release Date : 2006-01-18

Initiation To Global Finslerian Geometry written by Hassan Akbar-Zadeh and has been published by Elsevier this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-01-18 with Mathematics categories.

After a brief description of the evolution of thinking on Finslerian geometry starting from Riemann, Finsler, Berwald and Elie Cartan, the book gives a clear and precise treatment of this geometry. The first three chapters develop the basic notions and methods, introduced by the author, to reach the global problems in Finslerian Geometry. The next five chapters are independent of each other, and deal with among others the geometry of generalized Einstein manifolds, the classification of Finslerian manifolds of constant sectional curvatures. They also give a treatment of isometric, affine, projective and conformal vector fields on the unitary tangent fibre bundle.Key features- Theory of connections of vectors and directions on the unitary tangent fibre bundle.- Complete list of Bianchi identities for a regular conection of directions.- Geometry of generalized Einstein manifolds.- Classification of Finslerian manifolds.- Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle. - Theory of connections of vectors and directions on the unitary tangent fibre bundle. - Complete list of Bianchi identities for a regular conection of directions. - Geometry of generalized Einstein manifolds. - Classification of Finslerian manifolds. - Affine, isometric, conformal and projective vector fields on the unitary tangent fibre bundle.