Aspects Of Differential Geometry Ii

Author : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 143 pages
File Size : 50,6 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 : Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 140 pages
File Size : 55,6 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 : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool Publishers
Page : 169 pages
File Size : 53,8 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 : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Springer Nature
Page : 140 pages
File Size : 48,8 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 : Esteban Calviño-Louzao,Eduardo García-Río,Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenzo
Publisher : Morgan & Claypool Publishers
Page : 169 pages
File Size : 51,5 Mb
Release : 2017-05-25
Category : Mathematics
ISBN : 9781627058827

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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 : Richard S. Millman,George D. Parker
Publisher : Prentice Hall
Page : 288 pages
File Size : 48,8 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 : Chris J Isham
Publisher : World Scientific Publishing Company
Page : 304 pages
File Size : 44,5 Mb
Release : 1999-03-19
Category : Science
ISBN : 9789813102965

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Modern Differential Geometry for Physicists by Chris J Isham Pdf

This edition of the invaluable text Modern Differential Geometry for Physicists contains an additional chapter that introduces some of the basic ideas of general topology needed in differential geometry. A number of small corrections and additions have also been made. These lecture notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course “Quantum Fields and Fundamental Forces” at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen bearing in mind the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, nonlinear sigma models and other types of nonlinear field systems that feature in modern quantum field theory. The volume is divided into four parts: (i) introduction to general topology; (ii) introductory coordinate-free differential geometry; (iii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds; (iv) introduction to the theory of fibre bundles. In the introduction to differential geometry the author lays considerable stress on the basic ideas of “tangent space structure”, which he develops from several different points of view — some geometrical, others more algebraic. This is done with awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry.

Author : Shoshichi Kobayashi,Katsumi Nomizu
Publisher : Unknown
Page : 498 pages
File Size : 54,9 Mb
Release : 1963
Category : Mathematics
ISBN : STANFORD:36105030589191

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Foundations of Differential Geometry, Volume 2 by Shoshichi Kobayashi,Katsumi Nomizu Pdf

This two-volume introduction to differential geometry, part of Wiley's popular Classics Library, lays the foundation for understanding an area of study that has become vital to contemporary mathematics. It is completely self-contained and will serve as a reference as well as a teaching guide. Volume 1 presents a systematic introduction to the field from a brief survey of differentiable manifolds, Lie groups and fibre bundles to the extension of local transformations and Riemannian connections. The second volume continues with the study of variational problems on geodesics through differential geometric aspects of characteristic classes. Both volumes familiarize readers with basic computational techniques.

Author : Anonim
Publisher : Unknown
Page : 128 pages
File Size : 50,6 Mb
Release : 2024-06-26
Category : Electronic
ISBN : 9789814474771

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Lectures on the Geometry of Manifolds by Anonim Pdf

Author : Joel W. Robbin,Dietmar A. Salamon
Publisher : Springer Nature
Page : 426 pages
File Size : 49,9 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 : C J Isham
Publisher : World Scientific
Page : 192 pages
File Size : 43,9 Mb
Release : 1989-08-01
Category : Electronic
ISBN : 9789814507288

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Modern Differential Geometry for Physicists by C J Isham Pdf

These notes are the content of an introductory course on modern, coordinate-free differential geometry which is taken by the first-year theoretical physics PhD students, or by students attending the one-year MSc course “Fundamental Fields and Forces” at Imperial College. The book is concerned entirely with mathematics proper, although the emphasis and detailed topics have been chosen with an eye to the way in which differential geometry is applied these days to modern theoretical physics. This includes not only the traditional area of general relativity but also the theory of Yang-Mills fields, non-linear sigma-models and other types of non-linear field systems that feature in modern quantum field theory. This volume is in three parts dealing with, respectively, (i) introductory coordinate-free differential geometry, (ii) geometrical aspects of the theory of Lie groups and Lie group actions on manifolds, (iii) introduction to the theory of fibre bundles. In the first part of the book the author has laid considerable stress on the basic ideas of “tangent space structure” which he develops from several different points of view: some geometrical, and others more algebraic. This is done with the awareness of the difficulty which physics graduate students often experience when being exposed for the first time to the rather abstract ideas of differential geometry. Contents:An Introduction to TopologyDifferentiable ManifoldsVector Fields and n-FormsLie GroupsFibre BundlesConnections in a Bundle

Author : C. E. Weatherburn
Publisher : Cambridge University Press
Page : 253 pages
File Size : 41,6 Mb
Release : 1927
Category : Mathematics
ISBN : 9781316606957

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Differential Geometry of Three Dimensions by C. E. Weatherburn Pdf

Originally published in 1930, as the second of a two-part set, this textbook contains a vectorial treatment of geometry.

Author : Liviu I. Nicolaescu
Publisher : World Scientific
Page : 606 pages
File Size : 48,6 Mb
Release : 2007
Category : Mathematics
ISBN : 9789812708533

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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 W. Michor
Publisher : American Mathematical Soc.
Page : 510 pages
File Size : 46,8 Mb
Release : 2008
Category : Geometry, Differential
ISBN : 9780821820032

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Topics in Differential Geometry by Peter W. Michor Pdf

"This book treats the fundamentals of differential geometry: manifolds, flows, Lie groups and their actions, invariant theory, differential forms and de Rham cohomology, bundles and connections, Riemann manifolds, isometric actions, and symplectic and Poisson geometry. It gives the careful reader working knowledge in a wide range of topics of modern coordinate-free differential geometry in not too many pages. A prerequisite for using this book is a good knowledge of undergraduate analysis and linear algebra."--BOOK JACKET.

Author : Fangyang Zheng
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 46,7 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.