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Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications

Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications
  • Author : Bayram Sahin
  • Publisher :Unknown
  • Release Date :2017-01-23
  • Total pages :360
  • ISBN : 9780128044100
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Summary : Riemannian Submersions, Riemannian Maps in Hermitian Geometry, and their Applications is a rich and self-contained exposition of recent developments in Riemannian submersions and maps relevant to complex geometry, focusing particularly on novel submersions, Hermitian manifolds, and K\{a}hlerian manifolds. Riemannian submersions have long been an effective tool to obtain new manifolds and compare certain manifolds within differential geometry. For complex cases, only holomorphic submersions function appropriately, as discussed at length in Falcitelli, Ianus and Pastore’s classic 2004 book. In this new book, Bayram Sahin extends the scope of complex cases with wholly new submersion types, including Anti-invariant submersions, Semi-invariant submersions, slant submersions, and Pointwise slant submersions, also extending their use in Riemannian maps. The work obtains new properties of the domain and target manifolds and investigates the harmonicity and geodesicity conditions for such maps. It also relates these maps with discoveries in pseudo-harmonic maps. Results included in this volume should stimulate future research on Riemannian submersions and Riemannian maps. Systematically reviews and references modern literature in Riemannian maps Provides rigorous mathematical theory with applications Presented in an accessible reading style with motivating examples that help the reader rapidly progress

Riemannian Submersions and Related Topics

Riemannian Submersions and Related Topics
  • Author : Maria Falcitelli,Anna Maria Pastore,Stere Ianus?
  • Publisher :Unknown
  • Release Date :2004
  • Total pages :277
  • ISBN : 9789812562333
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Summary : This book provides the first-ever systematic introduction to thetheory of Riemannian submersions, which was initiated by BarrettO''Neill and Alfred Gray less than four decades ago. The authorsfocus their attention on classification theorems when the total spaceand the fibres have nice geometric properties.

Differential Geometry of Lightlike Submanifolds

Differential Geometry of Lightlike Submanifolds
  • Author : Krishan L. Duggal,Bayram Sahin
  • Publisher :Unknown
  • Release Date :2011-02-02
  • Total pages :488
  • ISBN : 3034602510
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Summary : This book presents research on the latest developments in differential geometry of lightlike (degenerate) subspaces. The main focus is on hypersurfaces and a variety of submanifolds of indefinite Kählerian, Sasakian and quaternion Kähler manifolds.

Manifolds II

Manifolds II
  • Author : Paul Bracken
  • Publisher :Unknown
  • Release Date :2019-05-22
  • Total pages :146
  • ISBN : 9781838803094
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Summary : Differential geometry is a very active field of research and has many applications to areas such as physics, in particular gravity. The chapters in this book cover a number of subjects that will be of interest to workers in these areas. It is hoped that these chapters will be able to provide a useful resource for researchers with regard to current fields of research in this important area.

Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications

Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications
  • Author : Krishan L. Duggal,Aurel Bejancu
  • Publisher :Unknown
  • Release Date :2013-04-17
  • Total pages :303
  • ISBN : 9789401720892
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Summary : This book is about the light like (degenerate) geometry of submanifolds needed to fill a gap in the general theory of submanifolds. The growing importance of light like hypersurfaces in mathematical physics, in particular their extensive use in relativity, and very limited information available on the general theory of lightlike submanifolds, motivated the present authors, in 1990, to do collaborative research on the subject matter of this book. Based on a series of author's papers (Bejancu [3], Bejancu-Duggal [1,3], Dug gal [13], Duggal-Bejancu [1,2,3]) and several other researchers, this volume was conceived and developed during the Fall '91 and Fall '94 visits of Bejancu to the University of Windsor, Canada. The primary difference between the lightlike submanifold and that of its non degenerate counterpart arises due to the fact that in the first case, the normal vector bundle intersects with the tangent bundle of the submanifold. Thus, one fails to use, in the usual way, the theory of non-degenerate submanifolds (cf. Chen [1]) to define the induced geometric objects (such as linear connection, second fundamental form, Gauss and Weingarten equations) on the light like submanifold. Some work is known on null hypersurfaces and degenerate submanifolds (see an up-to-date list of references on pages 138 and 140 respectively). Our approach, in this book, has the following outstanding features: (a) It is the first-ever attempt of an up-to-date information on null curves, lightlike hypersur faces and submanifolds, consistent with the theory of non-degenerate submanifolds.

Pseudo-Riemannian Geometry, δ-Invariants and Applications

Pseudo-Riemannian Geometry, δ-Invariants and Applications
  • Author : Bang-Yen Chen
  • Publisher :Unknown
  • Release Date :2011-03-23
  • Total pages :512
  • ISBN : 9789814462488
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Summary : The first part of this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian manifolds and their non-degenerate submanifolds, only assuming from the reader some basic knowledge about manifold theory. A number of recent results on pseudo-Riemannian submanifolds are also included. The second part of this book is on δ-invariants, which was introduced in the early 1990s by the author. The famous Nash embedding theorem published in 1956 was aimed for, in the hope that if Riemannian manifolds could be regarded as Riemannian submanifolds, this would then yield the opportunity to use extrinsic help. However, this hope had not been materialized as pointed out by M Gromov in his 1985 article published in Asterisque. The main reason for this is the lack of control of the extrinsic invariants of the submanifolds by known intrinsic invariants. In order to overcome such difficulties, as well as to provide answers for an open question on minimal immersions, the author introduced in the early 1990s new types of Riemannian invariants, known as δ-invariants, which are very different in nature from the classical Ricci and scalar curvatures. At the same time he was able to establish general optimal relations between δ-invariants and the main extrinsic invariants. Since then many new results concerning these δ-invariants have been obtained by many geometers. The second part of this book is to provide an extensive and comprehensive survey over this very active field of research done during the last two decades. Contents:Pseudo-Riemannian ManifoldsBasics on Pseudo-Riemannian SubmanifoldsSpecial Pseudo-Riemannian SubmanifoldsWarped Products and Twisted ProductsRobertson–Walker SpacetimesHodge Theory, Elliptic Differential Operators and Jacobi's Elliptic FunctionsSubmanifolds of Finite TypeTotal Mean CurvaturePseudo-Kähler ManifoldsPara-Kähler ManifoldsPseudo-Riemannian SubmersionsContact Metric Manifolds and Submanifoldsδ-Invariants, Inequalities and Ideal ImmersionsSome Applications of δ-InvariantsApplications to Kähler and Para-Kähler GeometryApplications to Contact GeometryApplications to Affine GeometryApplications to Riemannian SubmersionsNearly Kähler Manifolds and Nearly Kähler S6(1)δ(2)-Ideal Immersions Readership: Graduate and PhD students in differential geometry and related fields; researchers in differential geometry and related fields; theoretical physicists. Keywords:Pseudo-Riemannian Submanifold;δ-Invariants;Spacetimes;Submersion;Lagrangian Submanifolds;Sasakian Manifold;Total Mean Curvature;Submanifold of Finite Type;Affine HypersurfaceKey Features:This is the only book that provides general results on pseudo-Riemannian submanifoldsThis is the only book that provides detailed account on δ-invariantsAt the beginning of each chapter, historical background is providedReviews: “This book gives an extensive and in-depth overview of the theory of pseudo-Riemannian submanifolds and of the delta-invariants. It is written in an accessible and quite self-contained way. Hence it is recommendable for a very broad audience of students and mathematicians interested in the geometry of submanifolds.” Mathematical Reviews “This books is an extensive and comprehensive survey on pseudo–Riemannian submanifolds and δ–invariants as well as their applications. In every aspect, this is an excellent book, invaluable both for learning the topic and a reference. Therefore, it should be strongly recommended for students and mathematicians interested in the geometry of pseudo-Riemannian submanifolds.” Zentralblatt MATH

Structures on Manifolds

Structures on Manifolds
  • Author : K Yano,M Kon
  • Publisher :Unknown
  • Release Date :1985-02-01
  • Total pages :520
  • ISBN : 9789814602808
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Summary : Contents: Riemannian ManifoldsSubmanifolds of Riemannian ManifoldsComplex ManifoldsSubmanifolds of Kaehlerian ManifoldsContact ManifoldsSubmanifolds of Sasakian Manifoldsf-StructuresProduct ManifoldsSubmersions Readership: Mathematicians. Keywords:Riemannian Manifold;Submanifold;Complex Manifold;Contact Manifold;Kaehlerian Manifold;Sasakian Manifold;Anti-Invariant Submanifold;CR Submanifold;Contact CR Submanifold;Submersion

An Introduction to Riemannian Geometry

An Introduction to Riemannian Geometry
  • Author : Leonor Godinho,José Natário
  • Publisher :Unknown
  • Release Date :2014-07-26
  • Total pages :467
  • ISBN : 9783319086668
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Summary : Unlike many other texts on differential geometry, this textbook also offers interesting applications to geometric mechanics and general relativity. The first part is a concise and self-contained introduction to the basics of manifolds, differential forms, metrics and curvature. The second part studies applications to mechanics and relativity including the proofs of the Hawking and Penrose singularity theorems. It can be independently used for one-semester courses in either of these subjects. The main ideas are illustrated and further developed by numerous examples and over 300 exercises. Detailed solutions are provided for many of these exercises, making An Introduction to Riemannian Geometry ideal for self-study.

Mathematical Reviews

Mathematical Reviews
  • Author : Anonim
  • Publisher :Unknown
  • Release Date :2004
  • Total pages :229
  • ISBN : UOM:39015059777659
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Summary :

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture

Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture
  • Author : Peter B. Gilkey,John V Leahy,JeongHyeong Park
  • Publisher :Unknown
  • Release Date :1999-07-27
  • Total pages :296
  • ISBN : 0849382777
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Summary : This cutting-edge, standard-setting text explores the spectral geometry of Riemannian submersions. Working for the most part with the form valued Laplacian in the class of smooth compact manifolds without boundary, the authors study the relationship-if any-between the spectrum of Dp on Y and Dp on Z, given that Dp is the p form valued Laplacian and pi: Z ® Y is a Riemannian submersion. After providing the necessary background, including basic differential geometry and a discussion of Laplace type operators, the authors address rigidity theorems. They establish conditions that ensure that the pull back of every eigenform on Y is an eigenform on Z so the eigenvalues do not change, then show that if a single eigensection is preserved, the eigenvalues do not change for the scalar or Bochner Laplacians. For the form valued Laplacian, they show that if an eigenform is preserved, then the corresponding eigenvalue can only increase. They generalize these results to the complex setting as well. However, the spinor setting is quite different. For a manifold with non-trivial boundary and imposed Neumann boundary conditions, the result is surprising-the eigenvalues can change. Although this is a relatively rare phenomenon, the authors give examples-a circle bundle or, more generally, a principal bundle with structure group G where the first cohomology group H1(G;R) is non trivial. They show similar results in the complex setting, show that eigenvalues can decrease in the spinor setting, and offer a list of unsolved problems in this area. Moving to some related topics involving questions of positive curvature, for the first time in mathematical literature the authors establish a link between the spectral geometry of Riemannian submersions and the Gromov-Lawson conjecture. Spectral Geometry, Riemannian Submersions, and the Gromov-Lawson Conjecture addresses a hot research area and promises to set a standard for the field. Researchers and applied mathematicians interested in mathematical physics and relativity will find this work both fascinating and important.

Lectures on Kähler Geometry

Lectures on Kähler Geometry
  • Author : Andrei Moroianu
  • Publisher :Unknown
  • Release Date :2007-03-29
  • Total pages :229
  • ISBN : 9781139463003
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Summary : Kähler geometry is a beautiful and intriguing area of mathematics, of substantial research interest to both mathematicians and physicists. This self-contained graduate text provides a concise and accessible introduction to the topic. The book begins with a review of basic differential geometry, before moving on to a description of complex manifolds and holomorphic vector bundles. Kähler manifolds are discussed from the point of view of Riemannian geometry, and Hodge and Dolbeault theories are outlined, together with a simple proof of the famous Kähler identities. The final part of the text studies several aspects of compact Kähler manifolds: the Calabi conjecture, Weitzenböck techniques, Calabi–Yau manifolds, and divisors. All sections of the book end with a series of exercises and students and researchers working in the fields of algebraic and differential geometry and theoretical physics will find that the book provides them with a sound understanding of this theory.

Einstein Manifolds

Einstein Manifolds
  • Author : Arthur L. Besse
  • Publisher :Unknown
  • Release Date :2007-12-03
  • Total pages :510
  • ISBN : 9783540741206
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Summary : Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. This is the first book which presents an overview of several striking results ensuing from the examination of Einstein’s equations in the context of Riemannian manifolds. Parts of the text can be used as an introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.

Riemannian Submersions and Related Topics

Riemannian Submersions and Related Topics
  • Author : Maria Falcitelli,Anna Maria Pastore,Stere Ianus?
  • Publisher :Unknown
  • Release Date :2004
  • Total pages :277
  • ISBN : 9789812562333
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Summary : This book provides the first-ever systematic introduction to thetheory of Riemannian submersions, which was initiated by BarrettO''Neill and Alfred Gray less than four decades ago. The authorsfocus their attention on classification theorems when the total spaceand the fibres have nice geometric properties.

Geometry of Submanifolds

Geometry of Submanifolds
  • Author : Bang-Yen Chen
  • Publisher :Unknown
  • Release Date :2019-06-12
  • Total pages :192
  • ISBN : 9780486832784
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Summary : 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.

Revue Roumaine de Mathématiques Pures Et Appliquées

Revue Roumaine de Mathématiques Pures Et Appliquées
  • Author : Anonim
  • Publisher :Unknown
  • Release Date :2007
  • Total pages :229
  • ISBN : UOM:39015072688552
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Summary :

Handbook of Global Analysis

Handbook of Global Analysis
  • Author : Demeter Krupka,David Saunders
  • Publisher :Unknown
  • Release Date :2011-08-11
  • Total pages :1244
  • ISBN : 0080556736
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Summary : This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis , dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics. This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry. - Comprehensive coverage of modern global analysis and geometrical mathematical physics - Written by world-experts in the field - Up-to-date contents

The Ricci Flow in Riemannian Geometry

The Ricci Flow in Riemannian Geometry
  • Author : Ben Andrews,Christopher Hopper
  • Publisher :Unknown
  • Release Date :2011
  • Total pages :296
  • ISBN : 9783642162855
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Summary : Focusing on Hamilton's Ricci flow, this volume begins with a detailed discussion of the required aspects of differential geometry. The discussion also includes existence and regularity theory, compactness theorems for Riemannian manifolds, and much more.

The Mathematical Heritage of C F Gauss

The Mathematical Heritage of C F Gauss
  • Author : George M Rassias
  • Publisher :Unknown
  • Release Date :1991-09-30
  • Total pages :918
  • ISBN : 9789814603799
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Summary : This volume is a collection of original and expository papers in the fields of Mathematics in which Gauss had made many fundamental discoveries. The contributors are all outstanding in their fields and the volume will be of great interest to all research mathematicians, research workers in the history of science, and graduate students in Mathematics and Mathematical Physics. Contents: Powers of 2, Continued Fractions, and the Class Number One Problem for Real Quadratic Fields Q(√d), with d ?1(mod 8) (R A Mollin & H C Williams)A Load Balanced Algorithm for the Calculation of the Polynomial Knot and Link Invariants (B Ewing & K C Millett)The Maxwell Condition in Friedmann Cosmology (C D Hill)Hyper-Kähler Metrics and Monopoles (H Pedersen)Formulas for Higher-Order Finite Expansions of Composite Maps (K P Rybakowski)Gauge-Natural Operators Transforming Connections to the Tangent Bundle (I Kolář)Returns Under Bounded Number of Iterations (Z S Kowalski)The Inequalities with a Small Number of Solutions (J H Evertse & K Gyóry)Factorization in Quaternion Orders Over Number Fields (D R Estes)and other papers Readership: Mathematicians and mathematical physicists.

Differential Geometry Of Warped Product Manifolds And Submanifolds

Differential Geometry Of Warped Product Manifolds And Submanifolds
  • Author : Chen Bang-yen
  • Publisher :Unknown
  • Release Date :2017-05-29
  • Total pages :516
  • ISBN : 9789813208940
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Summary : 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.

Lectures on Kähler Manifolds

Lectures on Kähler Manifolds
  • Author : Werner Ballmann
  • Publisher :Unknown
  • Release Date :2006
  • Total pages :172
  • ISBN : 3037190256
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Summary : These notes are based on lectures the author held at the University of Bonn and the Erwin-Schrodinger-Institute in Vienna. The aim is to give a thorough introduction to the theory of Kahler manifolds with special emphasis on the differential geometric side of Kahler geometry. Some familiarity with global analysis and partial differential equations is assumed, in particular in the part on the Calabi conjecture.

Optimization Algorithms on Matrix Manifolds

Optimization Algorithms on Matrix Manifolds
  • Author : P.-A. Absil,R. Mahony,R. Sepulchre
  • Publisher :Unknown
  • Release Date :2009-04-11
  • Total pages :240
  • ISBN : 1400830249
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Summary : Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.