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Geometry Of Submanifolds And Homogeneous Spaces


Geometry Of Submanifolds And Homogeneous Spaces
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Geometry Of Submanifolds And Homogeneous Spaces


Geometry Of Submanifolds And Homogeneous Spaces
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Author : Andreas Arvanitoyeorgos
language : en
Publisher: MDPI
Release Date : 2020-01-03

Geometry Of Submanifolds And Homogeneous Spaces written by Andreas Arvanitoyeorgos and has been published by MDPI this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-03 with Mathematics categories.


The present Special Issue of Symmetry is devoted to two important areas of global Riemannian geometry, namely submanifold theory and the geometry of Lie groups and homogeneous spaces. Submanifold theory originated from the classical geometry of curves and surfaces. Homogeneous spaces are manifolds that admit a transitive Lie group action, historically related to F. Klein's Erlangen Program and S. Lie's idea to use continuous symmetries in studying differential equations. In this Special Issue, we provide a collection of papers that not only reflect some of the latest advancements in both areas, but also highlight relations between them and the use of common techniques. Applications to other areas of mathematics are also considered.



Geometry Of Submanifolds And Homogeneous Spaces


Geometry Of Submanifolds And Homogeneous Spaces
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Author : Andreas Arvanitogeōrgos
language : en
Publisher:
Release Date : 2019

Geometry Of Submanifolds And Homogeneous Spaces written by Andreas Arvanitogeōrgos and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019 with categories.




Geometry Of Submanifolds


Geometry Of Submanifolds
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Author : Bang-Yen Chen
language : en
Publisher: Courier Dover Publications
Release Date : 2019-06-12

Geometry Of Submanifolds written by Bang-Yen Chen and has been published by Courier Dover Publications this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-06-12 with Mathematics categories.


The first two chapters of this frequently cited reference provide background material in Riemannian geometry and the theory of submanifolds. Subsequent chapters explore minimal submanifolds, submanifolds with parallel mean curvature vector, conformally flat manifolds, and umbilical manifolds. The final chapter discusses geometric inequalities of submanifolds, results in Morse theory and their applications, and total mean curvature of a submanifold. Suitable for graduate students and mathematicians in the area of classical and modern differential geometries, the treatment is largely self-contained. Problems sets conclude each chapter, and an extensive bibliography provides background for students wishing to conduct further research in this area. This new edition includes the author's corrections.



Submanifolds And Holonomy


Submanifolds And Holonomy
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Author : Jurgen Berndt
language : en
Publisher: CRC Press
Release Date : 2016-02-22

Submanifolds And Holonomy written by Jurgen Berndt and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-02-22 with Mathematics categories.


Submanifolds and Holonomy, Second Edition explores recent progress in the submanifold geometry of space forms, including new methods based on the holonomy of the normal connection. This second edition reflects many developments that have occurred since the publication of its popular predecessor.New to the Second EditionNew chapter on normal holonom



Homogeneous Structures On Riemannian Manifolds


Homogeneous Structures On Riemannian Manifolds
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Author : F Tricerri
language : en
Publisher:
Release Date : 2014-02-20

Homogeneous Structures On Riemannian Manifolds written by F Tricerri and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-02-20 with categories.


The central theme of this book is the theorem of Ambrose and Singer, which gives for a connected, complete and simply connected Riemannian manifold a necessary and sufficient condition for it to be homogeneous. This is a local condition which has to be satisfied at all points, and in this way it is a generalization of E. Cartan's method for symmetric spaces. The main aim of the authors is to use this theorem and representation theory to give a classification of homogeneous Riemannian structures on a manifold. There are eight classes, and some of these are discussed in detail. Using the constructive proof of Ambrose and Singer many examples are discussed with special attention to the natural correspondence between the homogeneous structure and the groups acting transitively and effectively as isometrics on the manifold.



The Geometry Of Curvature Homogeneous Pseudo Riemannian Manifolds


The Geometry Of Curvature Homogeneous Pseudo Riemannian Manifolds
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Author : Peter B. Gilkey
language : en
Publisher: World Scientific
Release Date : 2007

The Geometry Of Curvature Homogeneous Pseudo Riemannian Manifolds written by Peter B. Gilkey and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007 with Science categories.


"Pseudo-Riemannian geometry is an active research field not only in differential geometry but also in mathematical physics where the higher signature geometries play a role in brane theory. An essential reference tool for research mathematicians and physicists, this book also serves as a useful introduction to students entering this active and rapidly growing field. The author presents a comprehensive treatment of several aspects of pseudo-Riemannian geometry, including the spectral geometry of the curvature tensor, curvature homogeneity, and Stanilov-Tsankov-Videv theory."--BOOK JACKET.



Introduction To Differential Geometry


Introduction To Differential Geometry
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Author : Joel W. Robbin
language : en
Publisher: Springer Nature
Release Date : 2022-01-12

Introduction To Differential Geometry written by Joel W. Robbin and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-01-12 with Mathematics categories.


This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.



Smooth Manifolds


Smooth Manifolds
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Author : Claudio Gorodski
language : en
Publisher: Springer Nature
Release Date : 2020-08-01

Smooth Manifolds written by Claudio Gorodski and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-08-01 with Mathematics categories.


This concise and practical textbook presents the essence of the theory on smooth manifolds. A key concept in mathematics, smooth manifolds are ubiquitous: They appear as Riemannian manifolds in differential geometry; as space-times in general relativity; as phase spaces and energy levels in mechanics; as domains of definition of ODEs in dynamical systems; as Lie groups in algebra and geometry; and in many other areas. The book first presents the language of smooth manifolds, culminating with the Frobenius theorem, before discussing the language of tensors (which includes a presentation of the exterior derivative of differential forms). It then covers Lie groups and Lie algebras, briefly addressing homogeneous manifolds. Integration on manifolds, explanations of Stokes’ theorem and de Rham cohomology, and rudiments of differential topology complete this work. It also includes exercises throughout the text to help readers grasp the theory, as well as more advanced problems for challenge-oriented minds at the end of each chapter. Conceived for a one-semester course on Differentiable Manifolds and Lie Groups, which is offered by many graduate programs worldwide, it is a valuable resource for students and lecturers alike.



Differential Geometry Of Warped Product Manifolds And Submanifolds


Differential Geometry Of Warped Product Manifolds And Submanifolds
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Author : Bang-yen Chen
language : en
Publisher: World Scientific
Release Date : 2017-05-29

Differential Geometry Of Warped Product Manifolds And Submanifolds written by Bang-yen Chen and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-05-29 with Mathematics categories.


A warped product manifold is a Riemannian or pseudo-Riemannian manifold whose metric tensor can be decomposed into a Cartesian product of the y geometry and the x geometry — except that the x-part is warped, that is, it is rescaled by a scalar function of the other coordinates y. The notion of warped product manifolds plays very important roles not only in geometry but also in mathematical physics, especially in general relativity. In fact, many basic solutions of the Einstein field equations, including the Schwarzschild solution and the Robertson-Walker models, are warped product manifolds.The first part of this volume provides a self-contained and accessible introduction to the important subject of pseudo-Riemannian manifolds and submanifolds. The second part presents a detailed and up-to-date account on important results of warped product manifolds, including several important spacetimes such as Robertson-Walker's and Schwarzschild's.The famous John Nash's embedding theorem published in 1956 implies that every warped product manifold can be realized as a warped product submanifold in a suitable Euclidean space. The study of warped product submanifolds in various important ambient spaces from an extrinsic point of view was initiated by the author around the beginning of this century.The last part of this volume contains an extensive and comprehensive survey of numerous important results on the geometry of warped product submanifolds done during this century by many geometers.



Riemannian Manifolds


Riemannian Manifolds
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Author : John M. Lee
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-06

Riemannian Manifolds written by John M. Lee 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 2006-04-06 with Mathematics categories.


This book is designed as a textbook for a one-quarter or one-semester graduate course on Riemannian geometry, for students who are familiar with topological and differentiable manifolds. It focuses on developing an intimate acquaintance with the geometric meaning of curvature. In so doing, it introduces and demonstrates the uses of all the main technical tools needed for a careful study of Riemannian manifolds. The author has selected a set of topics that can reasonably be covered in ten to fifteen weeks, instead of making any attempt to provide an encyclopedic treatment of the subject. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing Riemannian geometry. It then introduces the Riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. From then on, all efforts are bent toward proving the four most fundamental theorems relating curvature and topology: the Gauss–Bonnet theorem (expressing the total curvature of a surface in term so fits topological type), the Cartan–Hadamard theorem (restricting the topology of manifolds of nonpositive curvature), Bonnet’s theorem (giving analogous restrictions on manifolds of strictly positive curvature), and a special case of the Cartan–Ambrose–Hicks theorem (characterizing manifolds of constant curvature). Many other results and techniques might reasonably claim a place in an introductory Riemannian geometry course, but could not be included due to time constraints.