Aspects Of Differential Geometry V

Author : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 140 pages
File Size : 55,9 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031024320

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Aspects of Differential Geometry V by Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry and establish the Kodaira Embedding Theorem.

Author : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool Publishers
Page : 156 pages
File Size : 51,5 Mb
Release : 2015-02-01
Category : Mathematics
ISBN : 9781627056632

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Aspects of Differential Geometry I by Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.

Author : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 145 pages
File Size : 44,9 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031024108

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Aspects of Differential Geometry III by Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book III is aimed at the first-year graduate level but is certainly accessible to advanced undergraduates. It deals with invariance theory and discusses invariants both of Weyl and not of Weyl type; the Chern‒Gauss‒Bonnet formula is treated from this point of view. Homothety homogeneity, local homogeneity, stability theorems, and Walker geometry are discussed. Ricci solitons are presented in the contexts of Riemannian, Lorentzian, and affine geometry.

Author : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,Jeonghyeong Park,Ramón Vázquez-Lorenzo,Steven G. Krantz
Publisher : Morgan & Claypool
Page : 0 pages
File Size : 52,6 Mb
Release : 2019
Category : Geometry, Differential
ISBN : 1681735652

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Aspects of Differential Geometry IV by Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,Jeonghyeong Park,Ramón Vázquez-Lorenzo,Steven G. Krantz Pdf

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 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.

Author : Esteban Calviño-Louzao,Eduardo García-Río,Peter B Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool Publishers
Page : 158 pages
File Size : 40,6 Mb
Release : 2021-04-06
Category : Mathematics
ISBN : 9781636391113

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Aspects of Differential Geometry V by Esteban Calviño-Louzao,Eduardo García-Río,Peter B Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo Pdf

Book V completes the discussion of the first four books by treating in some detail the analytic results in elliptic operator theory used previously. Chapters 16 and 17 provide a treatment of the techniques in Hilbert space, the Fourier transform, and elliptic operator theory necessary to establish the spectral decomposition theorem of a self-adjoint operator of Laplace type and to prove the Hodge Decomposition Theorem that was stated without proof in Book II. In Chapter 18, we treat the de Rham complex and the Dolbeault complex, and discuss spinors. In Chapter 19, we discuss complex geometry and establish the Kodaira Embedding Theorem.

Author : Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik
Publisher : Birkhäuser
Page : 471 pages
File Size : 53,6 Mb
Release : 2017-06-23
Category : Mathematics
ISBN : 9783319528427

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Analytic, Algebraic and Geometric Aspects of Differential Equations by Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik Pdf

This volume consists of invited lecture notes, survey papers and original research papers from the AAGADE school and conference held in Będlewo, Poland in September 2015. The contributions provide an overview of the current level of interaction between algebra, geometry and analysis and demonstrate the manifold aspects of the theory of ordinary and partial differential equations, while also pointing out the highly fruitful interrelations between those aspects. These interactions continue to yield new developments, not only in the theory of differential equations but also in several related areas of mathematics and physics such as differential geometry, representation theory, number theory and mathematical physics. The main goal of the volume is to introduce basic concepts, techniques, detailed and illustrative examples and theorems (in a manner suitable for non-specialists), and to present recent developments in the field, together with open problems for more advanced and experienced readers. It will be of interest to graduate students, early-career researchers and specialists in analysis, geometry, algebra and related areas, as well as anyone interested in learning new methods and techniques.

Author : Liviu I. Nicolaescu
Publisher : World Scientific
Page : 606 pages
File Size : 54,9 Mb
Release : 2007
Category : Mathematics
ISBN : 9789812778628

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Lectures on the Geometry of Manifolds by Liviu I. Nicolaescu Pdf

The goal of this book is to introduce the reader to some of the most frequently used techniques in modern global geometry. Suited to the beginning graduate student willing to specialize in this very challenging field, the necessary prerequisite is a good knowledge of several variables calculus, linear algebra and point-set topology.The book's guiding philosophy is, in the words of Newton, that ?in learning the sciences examples are of more use than precepts?. We support all the new concepts by examples and, whenever possible, we tried to present several facets of the same issue.While we present most of the local aspects of classical differential geometry, the book has a ?global and analytical bias?. We develop many algebraic-topological techniques in the special context of smooth manifolds such as Poincar‚ duality, Thom isomorphism, intersection theory, characteristic classes and the Gauss-;Bonnet theorem.We devoted quite a substantial part of the book to describing the analytic techniques which have played an increasingly important role during the past decades. Thus, the last part of the book discusses elliptic equations, including elliptic Lpand H”lder estimates, Fredholm theory, spectral theory, Hodge theory, and applications of these. The last chapter is an in-depth investigation of a very special, but fundamental class of elliptic operators, namely, the Dirac type operators.The second edition has many new examples and exercises, and an entirely new chapter on classical integral geometry where we describe some mathematical gems which, undeservedly, seem to have disappeared from the contemporary mathematical limelight.

Author : Peter B. Gilkey,Jeonghyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool
Page : 0 pages
File Size : 42,8 Mb
Release : 2015
Category : Geometry, Differential
ISBN : 1627057838

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Aspects of Differential Geometry II by Peter B. Gilkey,Jeonghyeong Park,Ramón Vázquez-Lorenzo Pdf

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. In Book I, we focus on preliminaries. Chapter 1 provides an introduction to multivariable calculus and treats the Inverse Function Theorem, Implicit Function Theorem, the theory of the Riemann Integral, and the Change of Variable Theorem. Chapter 2 treats smooth manifolds, the tangent and cotangent bundles, and Stokes' Theorem. Chapter 3 is an introduction to Riemannian geometry. The Levi-Civita connection is presented, geodesics introduced, the Jacobi operator is discussed, and the Gauss-Bonnet Theorem is proved. The material is appropriate for an undergraduate course in the subject. We have given some different proofs than those that are classically given and there is some new material in these volumes. For example, the treatment of the Chern-Gauss-Bonnet Theorem for pseudo-Riemannian manifolds with boundary is new.

Author : Maks A. Akivis,Vladislav V. Goldberg
Publisher : Springer Science & Business Media
Page : 272 pages
File Size : 50,9 Mb
Release : 2006-04-18
Category : Mathematics
ISBN : 9780387215112

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Differential Geometry of Varieties with Degenerate Gauss Maps by Maks A. Akivis,Vladislav V. Goldberg Pdf

This book surveys the differential geometry of varieties with degenerate Gauss maps, using moving frames and exterior differential forms as well as tensor methods. The authors illustrate the structure of varieties with degenerate Gauss maps, determine the singular points and singular varieties, find focal images and construct a classification of the varieties with degenerate Gauss maps.

Author : S.P. Novikov,A.T. Fomenko
Publisher : Springer
Page : 490 pages
File Size : 42,8 Mb
Release : 2013-01-09
Category : Mathematics
ISBN : 9401578966

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Basic Elements of Differential Geometry and Topology by S.P. Novikov,A.T. Fomenko Pdf

Author : Richard S. Millman,George D. Parker
Publisher : Prentice Hall
Page : 288 pages
File Size : 53,7 Mb
Release : 1977
Category : Mathematics
ISBN : UOM:39015059064181

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Elements of Differential Geometry by Richard S. Millman,George D. Parker Pdf

This text is intended for an advanced undergraduate (having taken linear algebra and multivariable calculus). It provides the necessary background for a more abstract course in differential geometry. The inclusion of diagrams is done without sacrificing the rigor of the material. For all readers interested in differential geometry.

Author : Victor V. Prasolov
Publisher : Springer Nature
Page : 278 pages
File Size : 44,5 Mb
Release : 2022-02-10
Category : Mathematics
ISBN : 9783030922498

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Differential Geometry by Victor V. Prasolov Pdf

This book combines the classical and contemporary approaches to differential geometry. An introduction to the Riemannian geometry of manifolds is preceded by a detailed discussion of properties of curves and surfaces. The chapter on the differential geometry of plane curves considers local and global properties of curves, evolutes and involutes, and affine and projective differential geometry. Various approaches to Gaussian curvature for surfaces are discussed. The curvature tensor, conjugate points, and the Laplace-Beltrami operator are first considered in detail for two-dimensional surfaces, which facilitates studying them in the many-dimensional case. A separate chapter is devoted to the differential geometry of Lie groups.

Author : Fangyang Zheng
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 46,5 Mb
Release : 2000
Category : Complex manifolds
ISBN : 9780821829608

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Complex Differential Geometry by Fangyang Zheng Pdf

Discusses the differential geometric aspects of complex manifolds. This work contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. It discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles. It also gives a brief account of the surface classification theory.

Author : Brian F. Doolin,Clyde F. Martin
Publisher : Courier Corporation
Page : 178 pages
File Size : 46,6 Mb
Release : 2012-01-01
Category : Mathematics
ISBN : 9780486488165

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Introduction to Differential Geometry for Engineers by Brian F. Doolin,Clyde F. Martin Pdf

This outstanding guide supplies important mathematical tools for diverse engineering applications, offering engineers the basic concepts and terminology of modern global differential geometry. Suitable for independent study as well as a supplementary text for advanced undergraduate and graduate courses, this volume also constitutes a valuable reference for control, systems, aeronautical, electrical, and mechanical engineers. The treatment's ideas are applied mainly as an introduction to the Lie theory of differential equations and to examine the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, vector fields, exterior algebra, and Lie algebras. An appendix reviews concepts related to vector calculus, including open and closed sets, compactness, continuity, and derivative.

Author : John Oprea
Publisher : American Mathematical Soc.
Page : 469 pages
File Size : 48,5 Mb
Release : 2019-02-06
Category : Electronic
ISBN : 9781470450502

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Differential Geometry and Its Applications by John Oprea Pdf

Differential Geometry and Its Applications studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. That mix of ideas offers students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. Differential Geometry and Its Applications emphasizes that this visualization goes hand in hand with understanding the mathematics behind the computer construction. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract.