Riemann 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.
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.
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.
Riemann Finsler Geometry
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Author : Shiing-shen Chern
language : en
Publisher: World Scientific Publishing Company
Release Date : 2005-05-10
Riemann Finsler Geometry written by Shiing-shen Chern and has been published by World Scientific Publishing Company this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-05-10 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 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.
A Sampler Of Riemann Finsler Geometry
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Author : David Dai-Wai Bao
language : en
Publisher: Cambridge University Press
Release Date : 2004-11
A Sampler Of Riemann Finsler Geometry written by David Dai-Wai Bao and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-11 with Mathematics categories.
These expository accounts treat issues related to volume, geodesics, curvature and mathematical biology, with instructive examples.
Riemannian Geometry In An Orthogonal Frame
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Author : Elie Cartan
language : en
Publisher: World Scientific
Release Date : 2001
Riemannian Geometry In An Orthogonal Frame written by Elie Cartan and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Mathematics categories.
Elie Cartan's book Geometry of Riemannian Manifolds (1928) was one of the best introductions to his methods. It was based on lectures given by the author at the Sorbonne in the academic year 1925-26. A modernized and extensively augmented edition appeared in 1946 (2nd printing, 1951, and 3rd printing, 1988). Cartan's lectures in 1926-27 were different -- he introduced exterior forms at the very beginning and used extensively orthonormal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. The lectures were translated into Russian in the book Riemannian Geometry in an Orthogonal Frame (1960). This book has many innovations, such as the notion of intrinsic normal differentiation and the Gaussian torsion of a submanifold in a Euclidean multidimensional space or in a space of constant curvature, an affine connection defined in a normal fiber bundle of a submanifold, etc. The only book of Elie Cartan that was not available in English, it has now been translated into English by Vladislav V Goldberg, the editor of the Russian edition.
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.
Minimal Submanifolds In Pseudo Riemannian Geometry
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Author : Henri Anciaux
language : en
Publisher: World Scientific
Release Date : 2010-11-02
Minimal Submanifolds In Pseudo Riemannian Geometry written by Henri Anciaux and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-11-02 with Mathematics categories.
Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemannian case.For the first time, this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian geometry, only assuming from the reader some basic knowledge about manifold theory. Several classical results, such as the Weierstrass representation formula for minimal surfaces, and the minimizing properties of complex submanifolds, are presented in full generality without sacrificing the clarity of exposition. Finally, a number of very recent results on the subject, including the classification of equivariant minimal hypersurfaces in pseudo-Riemannian space forms and the characterization of minimal Lagrangian surfaces in some pseudo-Kähler manifolds are given.
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.