Aspects Of Differential Geometry I

Author : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool Publishers
Page : 156 pages
File Size : 41,5 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 : Morgan & Claypool Publishers
Page : 169 pages
File Size : 42,6 Mb
Release : 2019-04-18
Category : Mathematics
ISBN : 9781681735641

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Aspects of Differential Geometry IV by Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo 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 : Richard S. Millman,George D. Parker
Publisher : Prentice Hall
Page : 288 pages
File Size : 48,9 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 : Esteban Calviño-Louzao,Eduardo García-Río,Peter B Gilkey,Jeonghyeong Park,Ramón Vázquez-Lorenzo
Publisher : Unknown
Page : 156 pages
File Size : 44,5 Mb
Release : 2021-04-06
Category : Electronic
ISBN : 1636391109

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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 : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 140 pages
File Size : 40,9 Mb
Release : 2022-05-31
Category : Mathematics
ISBN : 9783031024078

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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. Table of Contents: Preface / Acknowledgments / Basic Notions and Concepts / Manifolds / Riemannian and Pseudo-Riemannian Geometry / Bibliography / Authors' Biographies / Index

Author : S.P. Novikov,A.T. Fomenko
Publisher : Springer Science & Business Media
Page : 500 pages
File Size : 54,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401578950

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

One service mathematics has rendered the 'Et moi ..., si j'avait su comment en revenir, je n'y serais point aile.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Matht"natics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics seNe as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series

Author : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 145 pages
File Size : 45,6 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 : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 143 pages
File Size : 51,7 Mb
Release : 2022-06-01
Category : Mathematics
ISBN : 9783031024085

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Aspects of Differential Geometry II 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. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship between simplicial cohomology, singular cohomology, sheaf cohomology, and de Rham cohomology is established. 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 total curvature and length of curves given by a single ODE is new as is the discussion of the total Gaussian curvature of a surface defined by a pair of ODEs.

Author : Galina Filipuk,Yoshishige Haraoka,Sławomir Michalik
Publisher : Birkhäuser
Page : 471 pages
File Size : 45,8 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 : Sorin Dragomir,Giuseppe Tomassini
Publisher : Springer Science & Business Media
Page : 499 pages
File Size : 48,6 Mb
Release : 2007-06-10
Category : Mathematics
ISBN : 9780817644833

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Differential Geometry and Analysis on CR Manifolds by Sorin Dragomir,Giuseppe Tomassini Pdf

Presents many major differential geometric acheivements in the theory of CR manifolds for the first time in book form Explains how certain results from analysis are employed in CR geometry Many examples and explicitly worked-out proofs of main geometric results in the first section of the book making it suitable as a graduate main course or seminar textbook Provides unproved statements and comments inspiring further study

Author : Fangyang Zheng
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 40,7 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 : Liviu I. Nicolaescu
Publisher : World Scientific
Page : 606 pages
File Size : 43,7 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 : Joel W. Robbin,Dietmar A. Salamon
Publisher : Springer Nature
Page : 426 pages
File Size : 42,5 Mb
Release : 2022-01-12
Category : Mathematics
ISBN : 9783662643402

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Introduction to Differential Geometry by Joel W. Robbin,Dietmar A. Salamon Pdf

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.

Author : Chris J. Isham
Publisher : Allied Publishers
Page : 308 pages
File Size : 42,6 Mb
Release : 2002
Category : Geometry, Differential
ISBN : 8177643169

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Modern Differential Geometry for Physicists by Chris J. Isham Pdf

Author : Loring W. Tu
Publisher : Springer
Page : 347 pages
File Size : 48,7 Mb
Release : 2017-06-01
Category : Mathematics
ISBN : 9783319550848

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Differential Geometry by Loring W. Tu Pdf

This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establish common notations, Appendix A recalls the basics of manifold theory. Additionally, in an attempt to make the exposition more self-contained, sections on algebraic constructions such as the tensor product and the exterior power are included. Differential geometry, as its name implies, is the study of geometry using differential calculus. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein's general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is also useful in topology, several complex variables, algebraic geometry, complex manifolds, and dynamical systems, among other fields. The field has even found applications to group theory as in Gromov's work and to probability theory as in Diaconis's work. It is not too far-fetched to argue that differential geometry should be in every mathematician's arsenal.