Differential and Riemannian geometry Books

Book SynopsisThis book covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. It treats in detail classical results on the relations between curvature and topology. The book features numerous exercises with full solutions and a series of detailed examples are picked up repeatedly to illustrate each new definition or property introduced.Trade ReviewFrom the reviews of the third edition: "This new edition maintains the clear written style of the original, including many illustrations … examples and exercises (most with solutions)." (Joseph E. Borzellino, Mathematical Reviews, 2005) "This book based on graduate course on Riemannian geometry … covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results … are treated in detail. … contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics … have been added and worked out in the same spirit." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004) "This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris. … Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples." (EMS Newsletter, December 2005) "The guiding line of this by now classic introduction to Riemannian geometry is an in-depth study of each newly introduced concept on the basis of a number of reoccurring well-chosen examples … . The book continues to be an excellent choice for an introduction to the central ideas of Riemannian geometry." (M. Kunzinger, Monatshefte für Mathematik, Vol. 147 (1), 2006)Table of Contents1 Differential manifolds.- 1.A From submanifolds to abstract manifolds.- 1.A.1 Submanifolds of Euclidean spaces.- 1.A.2 Abstract manifolds.- 1.A.3 Smooth maps.- 1.B The tangent bundle.- 1.B.1 Tangent space to a submanifold of Rn+k.- 1.B.2 The manifold of tangent vectors.- 1.B.3 Vector bundles.- 1.B.4 Tangent map.- 1.C Vector fields.- 1.C.1 Definitions.- 1.C.2 Another definition for the tangent space.- 1.C.3 Integral curves and flow of a vector field.- 1.C.4 Image of a vector field by a diffeomorphism.- 1.D Baby Lie groups.- 1.D.1 Definitions.- 1.D.2 Adjoint representation.- 1.E Covering maps and fibrations.- 1.E.1 Covering maps and quotients by a discrete group.- 1.E.2 Submersions and fibrations.- 1.E.3 Homogeneous spaces.- 1.F Tensors.- 1.F.1 Tensor product (a digest).- 1.F.2 Tensor bundles.- 1.F.3 Operations on tensors.- 1.F.4 Lie derivatives.- 1.F.5 Local operators, differential operators.- 1.F.6 A characterization for tensors.- 1.G. Differential forms.- 1.G.1 Definitions.- 1.G.2 Exterior derivative.- 1.G.3 Volume forms.- 1.G.4 Integration on an oriented manifold.- 1.G.5 Haar measure on a Lie group.- 1.H Partitions of unity.- 2 Riemannian metrics.- 2.A Existence theorems and first examples.- 2.A.1 Basic definitions.- 2.A.2 Submanifolds of Euclidean or Minkowski spaces.- 2.A.3 Riemannian submanifolds, Riemannian products.- 2.A.4 Riemannian covering maps, flat tori.- 2.A.5 Riemannian submersions, complex projective space.- 2.A.6 Homogeneous Riemannian spaces.- 2.B Covariant derivative.- 2.B.1 Connections.- 2.B.2 Canonical connection of a Riemannian submanifold.- 2.B.3 Extension of the covariant derivative to tensors.- 2.B.4 Covariant derivative along a curve.- 2.B.5 Parallel transport.- 2.B.6 natural metric on the tangent bundle.- 2.C Geodesies.- 2.C.1 Definition, first examples.- 2.C.2 Local existence and uniqueness for geodesies, exponential map.- 2.C.3 Riemannian manifolds as metric spaces.- 2.C.4 An invitation to isosystolic inequalities.- 2.C.5 Complete Riemannian manifolds, Hopf-Rinow theorem.- 2.C.6 Geodesies and submersions, geodesies of PnC.- 2.C.7 Cut-locus.- 2.C.8 The geodesic flow.- 2.D A glance at pseudo-Riemannian manifolds.- 2.D.1 What remains true?.- 2.D.2 Space, time and light-like curves.- 2.D.3 Lorentzian analogs of Euclidean spaces, spheres and hegeode spaces.- 2.D.4 (In)completeness.- 2.D.5 The Schwarzschild model.- 2.D.6 Hyperbolicity versus ellipticity.- 3 Curvature.- 3.A. The curvature tensor.- 3.A.1 Second covariant derivative.- 3.A.2 Algebraic properties of the curvature tensor.- 3.A.3 Computation of curvature: some examples.- 3.A.4 Ricci curvature, scalar curvature.- 3.B. First and second variation.- 3.B.1 Technical preliminaries.- 3.B.2 First variation formula.- 3.B.3 Second variation formula.- 3.C. Jacobi vector fields.- 3.C.1 Basic topics about second derivatives.- 3.C.2 Index form.- 3.C.3 Jacobi fields and exponential map.- 3.C.4 Applications.- 3.D. Riemannian submersions and curvature.- 3.D.1 Riemannian submersions and connections.- 3.D.2 Jacobi fields of PnC.- 3.D.3 O’Neill’s formula.- 3.D.4 Curvature and length of small circles. Application to Riemannian submersions.- 3.E. The behavior of length and energy in the neighborhood of a geodesic.- 3.E.1 Gauss lemma.- 3.E.2 Conjugate points.- 3.E.3 Some properties of the cut-locus.- 3.F Manifolds with constant sectional curvature.- 3.G Topology and curvature: two basic results.- 3.G.1 Myers’ theorem.- 3.G.2 Cartan-Hadamard’s theorem.- 3.H. Curvature and volume.- 3.H.1 Densities on a differentiable manifold.- 3.H.2 Canonical measure of a Riemannian manifold.- 3.H.3 Examples: spheres, hyperbolic spaces, complex projective spaces.- 3.H.4 Small balls and scalar curvature.- 3.H.5 Volume estimates.- 3.I. Curvature and growth of the fundamental group.- 3.I.1 Growth of finite type groups.- 3.I.2 Growth of the fundamental group of compact manifolds with negative curvature.- 3.J. Curvature and topology: some important results.- 3.J.1 Integral formulas.- 3.J.2 (Geo)metric methods.- 3.J.3 Analytic methods.- 3.J.4 Coarse point of view: compactness theorems.- 3.K. Curvature tensors and representations of the orthogonal group.- 3.K.1 Decomposition of the space of curvature tensors.- 3.K.2 Conformally flat manifolds.- 3.K.3 The Second Bianchi identity.- 3.L. Hyperbolic geometry.- 3.L.1 Introduction.- 3.L.2 Angles and distances in the hyperbolic plane.- 3.L.3 Polygons with “many” right angles.- 3.L.4 Compact surfaces.- 3.L.5 Hyperbolic trigonometry.- 3.L.6 Prescribing constant negative curvature.- 3.L.7 A few words about higher dimension.- 3.M. Conformai geometry.- 3.M.2 Introduction.- 3.M.3 The Möbius group.- 3.M.4 Conformai, elliptic and hyperbolic geometry.- 4 Analysis on manifolds.- 4.A. Manifolds with boundary.- 4.A.1 Introduction.- 4.A.2 Stokes theorem and integration by parts.- 4.B. Bishop inequality.- 4.B.1 Some commutation formulas.- 4.B.2 Laplacian of the distance function.- 4.B.3 Another proof of Bishop’s inequality.- 4.B.4 Heintze-Karcher inequality.- 4.C. Differential forms and cohomology.- 4.C.1 The de Rham complex.- 4.C.2 Differential operators and their formal adjoints.- 4.C.3 The Hodge-de Rham theorem.- 4.C.4 A second visit to the Bochner method.- 4.D. Basic spectral geometry.- 4.D.1 The Laplace operator and the wave equation.- 4.D.2 Statement of basic results on the spectrum.- 4.E. Some examples of spectra.- 4.E.1 Introduction.- 4.E.2 The spectrum of flat tori.- 4.E.3 Spectrum of (Sn, can).- 4.F The minimax principle.- 4.G Eigenvalues estimates.- 4.G.1 Introduction.- 4.G.2 Bishop’s inequality and coarse estimates.- 4.G.3 Some consequences of Bishop’s theorem.- 4.G.4 Lower bounds for the first eigenvalue.- 4.H. Paul Levy’s isoperimetric inequality.- 4.H.1 The statement.- 4.H.2 The proof.- 5 Riemannian submanifolds.- 5.A. Curvature of submanifolds.- 5.A.1 Second fundamental form.- 5.A.2 Curvature of hypersurfaces.- 5.A.3 Application to explicit computations of curvatures.- 5.B Curvature and convexity.- 5.C Minimal surfaces.- 5.C.1 First results.- 5.C.2 Surfaces with constant mean curvature.- A Some extra problems.- B Solutions of exercises.- List of figures.

Read more less

Book SynopsisBryce DeWitt, a student of Nobel Laureate Julian Schwinger, was himself one of the towering figures in 20th century physics, particularly renowned for his seminal contributions to quantum field theory, numerical relativity and quantum gravity. In late 1971 DeWitt gave a course on gravitation at Stanford University, leaving almost 400 pages of detailed handwritten notes. Written with clarity and authority, and edited by his former student Steven Christensen, these timeless lecture notes, containing material or expositions not found in any other textbooks, are a gem to be discovered or re-discovered by anyone seriously interested in the study of gravitational physics.Trade ReviewFrom the reviews:“DeWitt’s lectures cover interesting and detailed material which is rarely found in other text books. It is a book for the advanced reader.” (Norbert Dragon, General Relativity and Gravitation, Vol. 44, 2012)Table of ContentsReview of the Uses of Invariants in Special Relativity.- Accelerated Motion in Special Relativity.- Realization of Continuous Groups.- Riemannian Manifolds.- The Free Particle Geodesics.- Weak Field Approximation. Newton`s Theory.- Ensembles of Particles.- Production of Gravitational Fields by Matter.- Conservation Laws.- Phenomenological Description of a Conservative Continuous Medium.- Solubility of the Einstein and Matter Equations.- Energy, Momentum and Stress in the Gravitational Field.- Measurement of Asymptotic Field.- The Electromagnetic Field.- Gravitational Waves.- Spinning Bodies.- Weak Field Gravitational Wave.- Stationary Spherically (or Rotationally) Symmetric Metric.- Kerr Metric Subcalculations.- Friedmann Cosmology.- Dynamical Equations and Diffeomorphisms.

Read more less

Book SynopsisThese notes approximately transcribe a 15-week course on symplectic geometry I taught at UC Berkeley in the Fall of 1997. The course at Berkeley was greatly inspired in content and style by Victor Guillemin, whose masterly teaching of beautiful courses on topics related to s- plectic geometry at MIT, I was lucky enough to experience as a graduate student. I am very thankful to him! That course also borrowed from the 1997 Park City summer courses on symplectic geometry and topology, and from many talks and discussions of the symplectic geometry group at MIT. Among the regular participants in the MIT - formal symplectic seminar 93-96, I would like to acknowledge the contributions of Allen Knutson, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa Prato, Eugene Lerman, Jonathan Weitsman, Lisa Jeffrey, Reyer Sjamaar, Shaun Martin, Stephanie Singer, Sue Tolman and, last but not least, Yael Karshon. Thanks to everyone sitting in Math 242 in the Fall of 1997 for all the c- ments they made, and especially to those who wrote notes on the basis of which I was better able to reconstruct what went on: Alexandru Scorpan, Ben Davis, David Martinez,DonBarkauskas,EzraMiller,HenriqueBursztyn,John-PeterLund,Laura De Marco, Olga Radko, Peter P? rib' ?k, Pieter Collins, Sarah Packman, Stephen Bigelow, Susan Harrington, Tolga Etgu .. and Yi Ma.Trade Review“I find this to be both the best introduction to symplectic geometry as well as a model for how to introduce any field of study. … one feels the hand of a master in the text’s homework sets: concrete, illustrative, and enhancing the material presented. … For an upper-level undergraduate or beginning graduate student, Lectures on Symplectic Geometry remains, in my opinion, an ideal starting point into an exciting, active and growing area of mathematics.” (Andrew McInerney, MAA Reviews, June, 2018)From the reviews of the first printing Over the years, there have been several books written to serve as an introduction to symplectic geometry and topology, […] The text under review here fits well within this tradition, providing a useful and effective synopsis of the basics of symplectic geometry and possibly serving as the springboard for a prospective researcher. The material covered here amounts to the "usual suspects" of symplectic geometry and topology. From an introductory chapter of symplectic forms and symplectic algebra, the book moves on to many of the subjects that serve as the basis for current research:symplectomorphisms, Lagrangian submanifolds, the Moser theorems, Darboux-Moser-Weinstein theory, almost complex structures, Kãhler structures, Hamiltonian mechanics, symplectic reduction, etc. The text is written in a clear, easy-to-follow style, that is most appropriate in mathematical sophistication for second-year graduate students; […]. This text had its origins in a 15-week course that the author taught at UC Berkeley. There are some nice passages where the author simply lists some known results and some well-known conjectures, much as one would expect to see in a good lecture on the same subject. Particularly eloquent is the author’s discussion of the compact examples and counterexamples of symplectic, almost complex, complex and Kähler manifolds. Throughout the text, she uses specific, well-chosen examples to illustrate the results. In the initial chapter, she provides a detailed section on the classical example of the syrnplectic structure of the cotangent bundle of a manifold. Showing a good sense of pedagogy, the author often leaves these examples as well-planned homework assignments at the end of some of the sections. […] In all of these cases, the author gives the reader a chance to illustrate and understand the interesting results of each section, rather than relegating the tedious but needed results to the reader. Mathematical Reviews 2002iTable of ContentsSymplectic Manifolds.- Symplectic Forms.- Symplectic Form on the Cotangent Bundle.- Symplectomorphisms.- Lagrangian Submanifolds.- Generating Functions.- Recurrence.- Local Forms.- Preparation for the Local Theory.- Moser Theorems.- Darboux-Moser-Weinstein Theory.- Weinstein Tubular Neighborhood Theorem.- Contact Manifolds.- Contact Forms.- Contact Dynamics.- Compatible Almost Complex Structures.- Almost Complex Structures.- Compatible Triples.- Dolbeault Theory.- Kähler Manifolds.- Complex Manifolds.- Kähler Forms.- Compact Kähler Manifolds.- Hamiltonian Mechanics.- Hamiltonian Vector Fields.- Variational Principles.- Legendre Transform.- Moment Maps.- Actions.- Hamiltonian Actions.- Symplectic Reduction.- The Marsden-Weinstein-Meyer Theorem.- Reduction.- Moment Maps Revisited.- Moment Map in Gauge Theory.- Existence and Uniqueness of Moment Maps.- Convexity.- Symplectic Toric Manifolds.- Classification of Symplectic Toric Manifolds.- Delzant Construction.- Duistermaat-Heckman Theorems.

Read more less

Book SynopsisThese notes consist of two parts: Selected in York 1) Geometry, New 1946, Topics University Notes Peter Lax. by Differential in the 2) Lectures on Stanford Geometry Large, 1956, Notes J.W. University by Gray. are here with no essential They reproduced change. Heinz was a mathematician who mathema- Hopf recognized important tical ideas and new mathematical cases. In the phenomena through special the central idea the of a or difficulty problem simplest background is becomes clear. in this fashion a crystal Doing geometry usually lead serious allows this to to - joy. Hopf's great insight approach for most of the in these notes have become the st- thematics, topics I will to mention a of further try ting-points important developments. few. It is clear from these notes that laid the on Hopf emphasis po- differential Most of the results in smooth differ- hedral geometry. whose is both t1al have understanding geometry polyhedral counterparts, works I wish to mention and recent important challenging. Among those of Robert on which is much in the Connelly rigidity, very spirit R. and in - of these notes (cf. Connelly, Conjectures questions open International of Mathematicians, H- of gidity, Proceedings Congress sinki vol. 1, 407-414) 1978, .Table of ContentsSelected Topics in Geometry.- The Euler Characteristic and Related Topics.- Selected Topics in Elementary Differential Geometry.- The Isoperimetric Inequality and Related Inequalities.- The Elementary Concept of Area and Volume.- Differential Geometry in the Large.- Differential Geometry of Surfaces in the Small.- Some General Remarks on Closed Surfaces in Differential Geometry.- The Total Curvature (Curvatura Inteqra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements.- Hadamard’s Characterization of the Ovaloids.- Closed Surfaces with Constant Gauss Curvature (Hilbert’s Method) — Generalizations and Problems — General Remarks on Weinqarten Surfaces.- General Closed Surfaces of Genus O with Constant Mean Curvature — Generalizations.- Simple Closed Surfaces (of Arbitrary Genus) with Constant Mean Curvature — Generalizations.- The Congruence Theorem for Ovaloids.- Singularities of Surfaces with Constant Negative Gauss Curvature.

Read more less

Book SynopsisThe volume contains the texts of the main talks delivered at the International Symposium on Complex Geometry and Analysis held in Pisa, May 23-27, 1988. The Symposium was organized on the occasion of the sixtieth birthday of Edoardo Vesentini. The aim of the lectures was to describe the present situation, the recent developments and research trends for several relevant topics in the field. The contributions are by distinguished mathematicians who have actively collaborated with the mathematical school in Pisa over the past thirty years.Table of ContentsHyperkähler manifolds.- Affine differential geometry and holomorphic curves.- The meromorphic continuation of Kloosterman-Selberg zeta functions.- Deformation of compact Riemann surfaces Y of genus p with distinguished points P 1 …, P m ? Y.- On moduli of vector bundles.- Quasiconformal mappings on CR manifolds.- On the stability of positive semigroups generated by operator matrices.- The levi problem on algebraic manifolds.- A Banach-Steinhaus theorem for weak and order continuous operators.- Fixed points of holomorphic mappings.

Read more less

Book SynopsisThis book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWSTrade ReviewFrom the reviews: "In this monumental work, Marcel Berger manages to survey large parts of present day Riemannian geometry. … the book offers a great opportunity to get a first impression of some part of Riemannian geometry, together with hints for further reading." (A.Cap, Monatshefte für Mathematik, Vol. 145 (4), 2005) "Riemannian geometry has become a vast subject, influencing, famously, the development of general relativity and, more recently, the classification of 3-manifolds by hyperbolic structures … . Marcel Berger’s book is an overview of this enormous subject. … Virtually everything is illustrated with clear and useful diagrams … . This is the sort of book one could dip into or refer to over a period of years." (Peter Giblin, The Mathematical Gazette, March, 2005) "Marcel Berger’s A Panoramic View of Riemannian Geometry is without doubt the most comprehensive, original and idiosyncratic treatise on differential geometry … . he manages to include the most up-to-date references on even the most classical topics that he presents, and he puts far greater emphasis on applications. … the book concludes with a massive and extremely useful bibliography of 1310 items." (Robert Osserman, SIAM Review, Vol. 47 (1), 2005) "This book of one of the main contributors to Riemannian geometry has as a first goal to give an overview to most of the living topics of the subject. … A second goal is to show how many intuitive geometric questions lead to Riemannian geometry in a natural way. In my opinion it is this way of explaining … which makes the book highly recommendable to students as well as to experienced geometers. Furthermore, this book is clearly an encyclopedia in Riemannian geometry … ." (F.Manhart, Internationale Mathematische Nachrichten, Issue 197, 2004) "This book is really a panorama. … the reading creates pleasure for the interested reader. … the book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." (Jürgen Eichhorn, Mathematical Reviews, 2004 h) "Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. … enables the reader to obtain a sweeping panoramic view of almost the entirety of the field." (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 49 (3-4), 2003)Table of Contents0. Vector fields, tensors 1. Tensor Riemannian duality, the connection and the curvature 2. The parallel transport 3. Absolute (Ricci) calculus, commutation formulas 4. Hodge and the Laplacian, Bochners technique 5. Generalizing Gauss-Bonnet, characteristic classes and C. GEOMETRIC MEASURE THEORY AND PSEUDO-HOLOMORPHIC B. HIGHER DIMENSIONS A.THE CASE OF SURFACES IN R3 C. various other bundles 3. Harmonic maps between Riemannian manifolds 4. Low dimensional Riemannian geometry 5. Some generalizations of Riemannian geometry 6. Gromov mm-spaces 7. Submanifolds B. Spinors A. Exterior differential forms (and some others) C. RICCI FLAT KÄHLER AND HYPERKÄHLER MANIFOLDS 6. Kählerian manifolds (Kähler metrics) Chapter XI : SOME OTHER IMPORTANT TOPICS 1. Non compact manifolds 2. Bundles over Riemannian manifolds B. QUATERNIONIC-KÄHLER MANIFOLDS A. G2 AND Spin(7) HIERRACHY : HOLONOMY GROUPS AND KÄHLER MANIFOLDS 1. Definitions and philosophy 2. Examples 3. General structure theorems 4. The classification result 5. The rare cases b. on a given compact manifolds : closures Chapter X : GLOBAL PARALLEL TRANSPORT AND ANOTHER RIEMANNIAN a. collapsing C. THE CASE OF RICCI CURVATURE 12. Compactness, convergence results 13. The set of all Riemannian structures : collapsing B. MORE FINITENESS THEOREMS A. CHEEGERs FINITENESS THEOREM 11. Finiteness results of all Riemannian structures third part : Finiteness, compactness, collapsing and the space D. NEGATIVE VERSUS NONPOSITIVE CURVATURE 10. The negative side : Ricci curvature C. VOLUMES, FUNDAMENTAL GROUP B. QUASI-ISOMETRIES A. INTRODUCTION E. POSSIBLE APPROACHES, LOOKING FOR THE FUTURE 7. Ricci curvature : positive, nonnegative and just below 8. The positive side : scalar curvature 9. The negative side : sectional curvature D. POSITIVITY OF THE CURVATURE OPERATOR C. THE NON-COMPACT CASE B. HOMOLOGY TYPE AND THE FUNDAMENTAL GROUP A. THE KNOWN EXAMPLES 6. The positive side : sectional curvature second part : Curvature of a given sign1. Introduction 2. The positive pinching 3. Pinching around zero 4. The negative pinching 5. Ricci curvature pinching first part : Pinching problems b. hierarchy of curvaturesa. hopfs urge d. the set of constants, ricci flat metrics 18. The Yamabe problem Chapter IX : from curvature to topology 0. Some history and structure of the chapter c. moduli b. uniqueness a. existence b. homogeneous spaces and others 14. Examples from Analysis I : the evolution Ricci flow 15. Examples from Analysis II : the Kähler case 16. The sporadic examples 17. Around existence and uniqueness a. symmetric spaces THIRD PART : EINSTEIN MANIFOLDS 12. Hilberts variational principle and great hopes 13. The examples from the geometric hierachy 10. The case of Min R d/2 when d=4 11. Summing up questions on MinVol and Min(R) d/2 b. the simplicial volume of gromov a. using integral formulas d. cheeger-rong examples 9. Some cases where MinVol > 0 , Min Rd/2 > 0 c. nilmanifolds and the converse : almost flat manifolds b. wallachs type examples a. s1 fibrations and more examples MinDiam = 0 MinVol, MinDiam 5. Definitions 6. The case of surfaces 7. Generalities, compactness, finiteness and equivalence 8.Cases where MinVol = Min R d/2 = 0 and SECOND PART : WHICH METRIC IS THE LESS CURVED : Min R d/2 , FIRST PART: PURE GEOMETRIC FUNCTIONALS 1. Systolic quotients 2. Counting periodic geodesics 3. The embolic volume 4. Diameter/Injectivity riemannian metric on a given compact manifold ? 0. Introduction and a possible scheme of attack c. the structure on a given Sd and KPn 19. Inverse problems II : conjugacy of geodesics flows Chapter VIII : the search for distinguished metrics : what is the best b. bott and samelson theorems a. definitions and the need to be careful are closed 14. The case of negative curvature 15. The case of nonpositive curvature 16. Entropies on various space forms 17. From Osserman to Lohkamp 18. Inverse problems I : manifolds all of whose geodesics b. the various notions of

Read more less

Book SynopsisAt the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results. PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book’s value as a most welcome reference text on this subject. Trade ReviewFrom the reviews:"The book is aimed to old and new problems of optimal transport. … This meticulous work is based on very large bibliography … that is converted into a very valuable monograph that presents many statements and theorems written specifically for this approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes." (Mihail Voicu, Zentralblatt MATH, Vol. 1156, 2009)“This book wins the challenge to give a new and broad perspective on the multifacet topic of the optimal mass transport. … Besides extensive and accurate references therein the reader will find comments on related questions barely touched upon in the main text as well as lively presentations on how ideas and results have developed. This book should prove useful both to the expert and to the beginner looking for a reference text on the subject.” (Dario Cordero Erausquin, Mathematical Reviews, Issue 2010 f)“The book is an in-depth, modern, clear exposition of the advanced theory of optimal transport, and it tries to put together in a unified way almost all the recent developments of the theory. … the book is extremely well written and very pleasant to read. … I strongly recommend this excellent book to every researcher or graduate student in the field of optimal transport. … of interest to many mathematicians in different areas, who are simply interested in having an overview of the subject.” (Alessio Figalli, Bulletin of the American Mathematical Society, Vol. 47 (4), February, 2010)Table of ContentsCouplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge—Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Read more less

Book SynopsisEinstein'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.Trade ReviewFrom the reviews: "[...] an efficient reference book for many fundamental techniques of Riemannian geometry. [...] despite its length, the reader will have no difficulty in getting the feel of its contents and discovering excellent examples of all interaction of geometry with partial differential equeations, topology, and Lie groups. Above all, the book provides a clear insight into the scope and diversity of problems posed by its title."S.M. Salamon in MathSciNet 1988 "It seemed likely to anyone who read the previous book by the same author, namely Manifolds all of whose geodesic are closed, that the present book would be one of the most important ever published on Riemannian geometry. This prophecy is indeed fulfilled."T.J. Wilmore in Bulletin of the London Mathematical Society 1987 "Einstein Manifolds is accordingly described as Besse’s second book … . there is no doubt that Einstein Manifolds is a magnificient work of mathematical scholarship. … It is truly a seminal work on an incomparably fascinating and important subject." (Michael Berg, MathDL, March, 2008) "The present book is intended to be a complete reference book. … The book under review serves several purposes. It is an efficient reference for many fundamental techniques of Riemannian geometry as well as excellent examples of the interaction of geometry with partial differential equations, topology and Lie groups. Certainly the monograph provides a clear insight into the scope and diversity of problems posed by its title." (Adela-Gabriela Mihai, Zentralblatt MATH, Vol. 1147, 2008)Table of ContentsBasic Material.- Basic Material (Continued): Kähler Manifolds.- Relativity.- Riemannian Functionals.- Ricci Curvature as a Partial Differential Equation.- Einstein Manifolds and Topology.- Homogeneous Riemannian Manifolds.- Compact Homogeneous Kähler Manifolds.- Riemannian Submersions.- Holonomy Groups.- Kähler-Einstein Metrics and the Calabi Conjecture.- The Moduli Space of Einstein Structures.- Self-Duality.- Quaternion-Kähler Manifolds.- A Report on the Non-Compact Case.- Generalizations of the Einstein Condition.

Read more less

Book SynopsisThis is comprehensive basic monograph on mixed Hodge structures. Building up from basic Hodge theory the book explains Delingne's mixed Hodge theory in a detailed fashion. Then both Hain's and Morgan's approaches to mixed Hodge theory related to homotopy theory are sketched. Next comes the relative theory, and then the all encompassing theory of mixed Hodge modules. The book is interlaced with chapters containing applications. Three large appendices complete the book.Trade ReviewFrom the reviews: "This book is dealing with Hodge Theory ... which generalizes in a functorial way the variations of MHS. ... The clarity of the presentation and the wealth of information are both remarkable. This book ... is a masterpiece that anyone working in Algebraic Geometry, Singularities or Analytic/Complex Geometry would like to have in his own library." (Alexandru Dimca, Zentralblatt MATH, Vol. 1138 (16), 2008) "The book under review … focuses mainly on the ‘pure’ story just summarized, is aimed at graduate students and researchers … . The book begins with a brief historical survey; each chapter is headed by a good summary of its contents and concluded by historical remarks (with references). … this work is a thoroughly readable and very up-to-date account of mixed Hodge theory, written by masters of the subject, and will undoubtedly serve as a basic reference for years to come." (Matt Kerr, Mathematical Reviews, Issue 2009 C) “This book has been awaited for many years. … the book which is now available will certainly rapidly become one of the standard references on the topic. Hodge theory assigns to a complex variety data which come from linear algebra. … I heartily recommend the book.” (Helene Esnault, Jahresbericht der Deutsche Mathematiker Vereinigung, Vol. 112 (1), 2010)Table of ContentsBasic Hodge Theory.- Compact Kähler Manifolds.- Pure Hodge Structures.- Abstract Aspects of Mixed Hodge Structures.- Mixed Hodge Structures on Cohomology Groups.- Smooth Varieties.- Singular Varieties.- Singular Varieties: Complementary Results.- Applications to Algebraic Cycles and to Singularities.- Mixed Hodge Structures on Homotopy Groups.- Hodge Theory and Iterated Integrals.- Hodge Theory and Minimal Models.- Hodge Structures and Local Systems.- Variations of Hodge Structure.- Degenerations of Hodge Structures.- Applications of Asymptotic Hodge Theory.- Perverse Sheaves and D-Modules.- Mixed Hodge Modules.

Read more less

Book SynopsisA collection of five surveys on dynamical systems, indispensable for graduate students and researchers in mathematics and theoretical physics. Written in the modern language of differential geometry, the book covers all the new differential geometric and Lie-algebraic methods currently used in the theory of integrable systems.Table of ContentsContents: Nonholonomic Dynamical Systems, Geometry of Distributions and Variational Problems by A.M. Vershik, V.Ya. Gershkovich.- Integrable Systems and Infinite Dimensional Lie Algebras by M.A. Olshanetsky, M.A. Perelomov.- Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems by A.G. Reyman, M.A. Semenov-Tian-Shansky.- Quantization of Open Toda Lattices by M.A. Semenov-Tian-Shansky.- Geometric and Algebraic Mechanisms of the Integrability of Hamiltonian Systems on Homogeneous Spaces and Lie Algebras by V.V. Trofimov, A.T. Fomenko.

Read more less

Book SynopsisThis book arose out of original research on the extension of well-established applications of complex numbers related to Euclidean geometry and to the space-time symmetry of two-dimensional Special Relativity. The system of hyperbolic numbers is extensively studied, and a plain exposition of space-time geometry and trigonometry is given. Commutative hypercomplex systems with four unities are studied and attention is drawn to their interesting properties.Trade ReviewFrom the reviews: “It is worth pointing out that the book is mainly a text about commutative hypercomplex numbers and some of their applications to a 2-dimensional Minkowski spacetime. … This book should be interesting to anybody who is interested in applications of hypercomplex numbers … . In conclusion, I recommend this book to anyone who wants to learn about hypercomplex numbers.” (Emanuel Gallo, Mathematical Reviews, Issue 2010 d)Table of ContentsThe Mathematics of Minkowski Space-Time: 1 N-Dimensional Hypercomplex Numbers and the associated Geometries.- Commutative Hypercomplex Number Systems.- The General Two-Dimensional System.- Linear Transformations and Geometries.- The Geometries Associated with Hypercomplex Numbers.- Conclusions.- 2 Trigonometry in the Minkowski Plane.- Geometrical Representation of Hyperbolic Numbers.- Basics of Hyperbolic Trigonometry.- Geometry in Pseudo-Euclidean Cartesian Plane.- Trigonometry in the Pseudo-Euclidean Plane.- Theorems on Equilateral Hyperbolas in the Pseudo-Euclidean Plane.- Some Examples of Triangle Solutions in the Minkowski Plane.- Conclusions.- 3 Uniform and Accelerated Motions in the Minkowski Space-Time (Twin Paradox).- Inertial Motions.- Inertial and Uniformly Accelerated Motions.- Non-uniformly Accelerated Motions.- Conclusions.- 4 General Two-Dimensional Hypercomplex Numbers.-Geometrical Representation.- Geometry and Trigonometry in Two-Dimensional Algebras.- Some Properties of Fundamental Conic Section.- Numerical Examples.- 5 Functions of a Hyperbolic Variable.- Some Remarks on Functions of a Complex Variable.- Functions of Hypercomplex Variables.- The Functions of a Hyperbolic Variable.- The Elementary Functions of a Canonical Hyperbolic Variable.- H-Conformal Mappings.- Commutative Hypercomplex Systems with Three Unities.- 6 Hyperbolic Variables on Lorentz Surfaces.- Introduction.- Gauss: Conformal Mapping of Surfaces.- Extension of Gauss Theorem: Conformal Mapping of Lorentz Surfaces.- Beltrami: Complex Variables on a Surface.- Beltrami’s Integration of Geodesic Equations.- Extension of Beltrami’s Equation to Non-Definite Differential Forms.- 7 Constant Curvature Lorentz Surfaces.- Introduction.- Constant Curvature RiemannSurfaces.- Constant Curvature Lorentz Surfaces.- Geodesics and Geodesic Distances on Riemann and Lorentz Surfaces.- Conclusions.- 8 Generalization of Two-Dimensional Special Relativity (Hyperbolic Transformations and the Equivalence Principle).- Physical Meaning of Transformations by Hyperbolic Functions.- Physical Interpretation of Geodesics on Riemann and Lorentz Surfaces with Positive Constant Curvature.- Einstein’s Way to General Relativity.- Conclusions.- II An Introduction to Commutative Hypercomplex Numbers.- 9 Commutative Segre’s Quaternions.- Introduction.- Hypercomplex Systems with Four Units.- Historical Introduction of Segre’s Quaternion.- Algebraic Properties of Commutative Quaternions.- Functions of a Quaternion Variable.- Mapping by Means of Quaternion Functions.- Elementary Functions of the Quaternions.- Elliptic-Hyperbolic Quaternions.- Elliptic-Parabolic Generalized Segre’s Quaternions.- 10 Constant Curvature Segre’s Quaternion Spaces.- Introduction.- Quaternion differential geometry and geodesic equations.- Orthogonality in Segre’s Quaternion Space.- Constant Curvature Quaternion Spaces.- Geodesic Equations in Quaternion Space.- Beltrami’s Integration Method for Quaternion Spaces.- Beltrami’s Integration Method for Quaternion Spaces.- Conclusions.- 11 A Matrix Formalization for Commutative Hypercomplex Systems.- Mathematical Operations.- Properties of the Characteristic Matrix M.- Functions of Hypercomplex Variable.- Functions of a Two-Dimensional Hypercomplex Variable.- Derivatives of a Hypercomplex Function.- Characteristic Differential Equation.- A Equivalence Between the Formalizations of Hypercomplex Numbers.

Read more less

Book SynopsisThe focus of this book is on providing students with insights into geometry that can help them understand deep learning from a unified perspective. Rather than describing deep learning as an implementation technique, as is usually the case in many existing deep learning books, here, deep learning is explained as an ultimate form of signal processing techniques that can be imagined. To support this claim, an overview of classical kernel machine learning approaches is presented, and their advantages and limitations are explained. Following a detailed explanation of the basic building blocks of deep neural networks from a biological and algorithmic point of view, the latest tools such as attention, normalization, Transformer, BERT, GPT-3, and others are described. Here, too, the focus is on the fact that in these heuristic approaches, there is an important, beautiful geometric structure behind the intuition that enables a systematic understanding. A unified geometric analysis to understand the working mechanism of deep learning from high-dimensional geometry is offered. Then, different forms of generative models like GAN, VAE, normalizing flows, optimal transport, and so on are described from a unified geometric perspective, showing that they actually come from statistical distance-minimization problems.Because this book contains up-to-date information from both a practical and theoretical point of view, it can be used as an advanced deep learning textbook in universities or as a reference source for researchers interested in acquiring the latest deep learning algorithms and their underlying principles. In addition, the book has been prepared for a codeshare course for both engineering and mathematics students, thus much of the content is interdisciplinary and will appeal to students from both disciplines.Trade Review“This book is based on material that has been prepared for senior-level undergraduate classes, this book can be used for one-semester senior-level undergraduate and graduate-level classes.” (Arzu Ahmadova, zbMATH 1493.68003, 2022)Table of ContentsPart I Basic Tools for Machine Learning: 1. Mathematical Preliminaries.- 2. Linear and Kernel Classifiers.- 3. Linear, Logistic, and Kernel Regression.- 4. Reproducing Kernel Hilbert Space, Representer Theorem.- Part II Building Blocks of Deep Learning: 5. Biological Neural Networks.- 6. Artificial Neural Networks and Backpropagation.- 7. Convolutional Neural Networks.- 8. Graph Neural Networks.- 9. Normalization and Attention.- Part III Advanced Topics in Deep Learning.- 10. Geometry of Deep Neural Networks.- 11. Deep Learning Optimization.- 12. Generalization Capability of Deep Learning.- 13. Generative Models and Unsupervised Learning.- Summary and Outlook.- Bibliography.- Index.

Read more less

Book SynopsisChapter 1 Riemannian Geometry.- Chapter 2 Finslerian Geometry.- Chapter 3 Isometric immersions.

Read more less

Book SynopsisPseudohermitian geometry.- CR manifolds with boundary.- Jacobi fields of the Tanaka-Webster connection.- CR immersions and Lorentzian geometry.- Proper holomorphic maps in harmonic map theory.- Beltrami equations on Rossi sphere.- CR immersions.

Read more less

a huge range and FREE tracked UK delivery on ALL orders.

Read more less

Book SynopsisThe latest volume in the New York Times–bestselling physics series explains Einstein’s masterpiece: the general theory of relativity He taught us classical mechanics, quantum mechanics, and special relativity. Now, physicist Leonard Susskind, assisted by a new collaborator, André Cabannes, returns to tackle Einstein’s general theory of relativity. Starting from the equivalence principle and covering the necessary mathematics of Riemannian spaces and tensor calculus, Susskind and Cabannes explain the link between gravity and geometry. They delve into black holes, establish Einstein field equations, and solve them to describe gravity waves. The authors provide vivid explanations that, to borrow a phrase from Einstein himself, are as simple as possible (but no simpler). An approachable yet rigorous introduction to one of the most important topics in physics, General Relativity is a must-read for anyone who wants a deeper knowledge of the universe’s real structure.  

Read more less

Book SynopsisUnlike 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.Trade ReviewFrom the book reviews:“The aim of the textbook is twofold. First, it is a concise and self-contained quick introduction to the basics of differential geometry, including differential forms, followed by the main ideas of Riemannian geometry. Second, the last two chapters are devoted to some interesting applications to geometric mechanics and relativity. … the book is well written and also very readable. I warmly recommend it to specialists in mathematics, physics and engineering, especially to Ph.D. students.” (Miroslaw Doupovec, zbMATH 1306.53001, 2015)Table of ContentsDifferentiable Manifolds.- Differential Forms.- Riemannian Manifolds.- Curvature.- Geometric Mechanics.- Relativity.

Read more less

Book SynopsisAn application of differential forms for the study of some local and global aspects of the differential geometry of surfaces. Differential forms are introduced in a simple way that will make them attractive to "users" of mathematics. A brief and elementary introduction to differentiable manifolds is given so that the main theorem, namely Stokes' theorem, can be presented in its natural setting. The applications consist in developing the method of moving frames expounded by E. Cartan to study the local differential geometry of immersed surfaces in R3 as well as the intrinsic geometry of surfaces. This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.Trade ReviewM.P. Do Carmo Differential Forms and Applications "This book treats differential forms and uses them to study some local and global aspects of differential geometry of surfaces. Each chapter is followed by interesting exercises. Thus, this is an ideal book for a one-semester course."—ACTA SCIENTIARUM MATHEMATICARUMTable of Contents1. Differential Forms in Rn.- 2. Line Integrals.- 3. Differentiable Manifolds.- 4. Integration on Manifolds; Stokes Theorem and Poincaré’s Lemma.- 1. Integration of Differential Forms.- 2. Stokes Theorem.- 3. Poincaré’s Lemma.- 5. Differential Geometry of Surfaces.- 1. The Structure Equations of Rn.- 2. Surfaces in R3.- 3. Intrinsic Geometry of Surfaces.- 6. The Theorem of Gauss-Bonnet and the Theorem of Morse.- 1. The Theorem of Gauss-Bonnet.- 2. The Theorem of Morse.- References.

Read more less

Book SynopsisDieses Buch ist eine Einführung in die Differentialgeometrie und ein passender Begleiter zum Differentialgeometrie-Modul (ein- und zweisemestrig). Zunächst geht es um die klassischen Aspekte wie die Geometrie von Kurven und Flächen, bevor dann höherdimensionale Flächen sowie abstrakte Mannigfaltigkeiten betrachtet werden. Die Nahtstelle ist dabei das zentrale Kapitel "Die innere Geometrie von Flächen". Dieses führt den Leser bis hin zu dem berühmten Satz von Gauß-Bonnet, der ein entscheidendes Bindeglied zwischen lokaler und globaler Geometrie darstellt. Die zweite Hälfte des Buches ist der Riemannschen Geometrie gewidmet. Den Abschluss bildet ein Kapitel über "Einstein-Räume", die eine große Bedeutung sowohl in der "Reinen Mathematik" als auch in der Allgemeinen Relativitätstheorie von A. Einstein haben. Es wird großer Wert auf Anschaulichkeit gelegt, was durch zahlreiche Abbildungen unterstützt wird. Bei der Neuauflage wurden einige zusätzliche Lösungen zu den Übungsaufgaben ergänzt.Table of ContentsBezeichnungen sowie Hilfsmittel aus der Analysis.- Kurven im IRn.- Lokale Flächentheorie, insbes. Drehflächen, Regelflächen, Minimalflächen.- Die innere Geometrie von Flächen.- Riemannsche Mannigfaltigkeiten.- Der Krümmungstensor.- Räume konstanter Krümmung.- Einstein-Räume.- Lösungen zu Übungsaufgaben.

Read more less

Book SynopsisWhat Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.Trade Review"This book is very well writen, pleasant to read, with many good illustrations. It deals with the core of the subject, nothing more and nothing less. Simply a recommendation for anyone who wants to teach or learn about the Riemannian geometry."Nieuw Archief voor Wiskunde, September 2000Table of ContentsWhat Is Curvature?.- Review of Tensors, Manifolds, and Vector Bundles.- Definitions and Examples of Riemannian Metrics.- Connections.- Riemannian Geodesics.- Geodesics and Distance.- Curvature.- Riemannian Submanifolds.- The Gauss-Bonnet Theorem.- Jacobi Fields.- Curvature and Topology.

Read more less

Book SynopsisComprehensive coverage of the foundations, applications, recent developments, and future of conformal differential geometry. Conformal Differential Geometry and Its Generalizations systematically presents the foundations and manifestations of conformal differential geometry.Table of ContentsConformal and Pseudoconformal Spaces. Hypersurfaces in Conformal Spaces. Submanifolds in Conformal and Pseudoconformal Spaces. Conformal Structures on a Differentiable Manifold. The Four-Dimensional Conformal Structures. Geometry of the Grassmann Manifold. Manifolds Endowed with Almost Grassmann Structures. Bibliography. Symbols Frequently Used. Indexes.

Read more less

Book SynopsisThis book describes integration and measure theory for readers interested in analysis, engineering, and economics. It gives a systematic account of Riemann-Stieltjes integration and deduces the Lebesgue-Stieltjes measure from the Lebesgue-Stieltjes integral.Table of ContentsLIMITATIONS OF THE RIEMANN INTEGRAL. Limits of Integrals and Integrability. Expectations in Probability Theory. RIEMANN-STIELTJES INTEGRALS. Riemann-Stieltjes Integrals: Introduction. Characterization of Riemann-Stieltjes Integrability. Continuous Linear Functionals on C[a,b]. Riemann-Stieltjes Integrals: Further Properties. LEBESGUE-STIELTJES INTEGRALS. The Extension of the Riemann-Stieltjes Integral. Lebesgue-Stieltjes Integrals. MEASURE THEORY. sigma-Algebras and Algebras of Sets. Measurable Functions. Measures. Lebesgue-Stieltjes Measures. THE ABSTRACT LEBESGUE INTEGRAL. The Integral Associated with a Measure Space. The Lebesgue Spaces and Norms. Absolutely Continuous Measures. Linear Functionals on the Lebesgue Spaces. Product Measures and Fubini's Theorem. Lebesgue Integration and Measures on R?n. Signed Measures and Complex Measures. Differentiation. Convergence of Sequences of Functions. Measures on Locally Compact Spaces. Hausdorff Measures and Dimension. Lorentz Spaces. Appendices. Indexes.

Read more less

Book SynopsisThe description for this book, Seminar on Atiyah-Singer Index Theorem. (AM-57), will be forthcoming.Table of Contents*Frontmatter, pg. i*CONTENTS, pg. v*PREFACE, pg. ix*CHAPTER I. STATEMENT OF THE THEOREM OUTLINE OF THE PROOF, pg. 1*CHAPTER II. REVIEW OF K-THEORY, pg. 13*CHAPTER III. THE TOPOLOGICAL INDEX OF AN OPERATOR ASSOCIATED TO A G-STRUCTURE, pg. 27*CHAPTER IV. DIFFERENTIAL OPERATORS ON VECTOR BUNDLES, pg. 51*CHAPTER V. ANALYTICAL INDICES OF SOME CONCRETE OPERATORS, pg. 95*CHAPTER VI. REVIEW OF FUNCTIONAL ANALYSIS, pg. 107*CHAPTER VII. FREDHDIM OPERATORS, pg. 119*CHAPTER VIII. CHAINS OP HILBERTIAN SPACES, pg. 125*CHAPTER IX. THE DISCRETE SOBOLEV CHAIN OF A VECTOR BUNDLE, pg. 147*CHAPTER X. THE CONTINUOUS SOBOLEV CHAIN OF A VECTOR BUNDLE, pg. 155*CHAPTER XI. THE SEELEY ALGEBRA, pg. 175*CHAPTER XII. HOMOTOPY INVARIANCE OF THE INDEX, pg. 185*CHAPTER XIII. WHITNEY SUMS, pg. 191*CHAPTER XIV. TENSOR PRODUCTS, pg. 197*CHAPTER XV. DEFINITION OF ia AND it ON K(M), pg. 215*CHAPTER XVI. CONSTRUCTION OF Intk, pg. 235*CHAPTER XVII. COBORDISM INVARIANCE OP THE ANALYTICAL INDEX, pg. 285*CHAPTER XVIII. BORDISM GROUPS OF BUNDLES, pg. 303*CHAPTER XIX. THE INDEX THEOREM: APPLICATIONS, pg. 313*APPENDIX I. THE INDEX THEOREM FOR MANIFOLDS WITH BOUNDARY, pg. 337*APPENDIX II. NON-STABLE CHARACTERISTIC CLASSES AND THE TOPOLOGICAL INDEX OP CLASSICAL ELLIPTIC OPERATORS, pg. 353*Backmatter, pg. 368

Read more less

Book SynopsisTrade Review"John Milnor, Winner of the 2011 Abel Prize from the Norwegian Academy of Science and Letters""John Willard Milnor, Winner of the 2011 Leroy P. Steele Prize for Lifetime Achievement, American Mathematical Society"Table of Contents*Frontmatter, pg. i*Preface, pg. v*Contents, pg. vii* 1. Smooth Manifolds, pg. 1* 2. Vector Bundles, pg. 13* 3. Constructing New Vector Bundles Out of Old, pg. 25* 4. Stiefel-Whitney Classes, pg. 37* 5. Grassmann Manifolds and Universal Bundles, pg. 55* 6. A Cell Structure for Grassmann Manifolds, pg. 73* 7. The Cohomology Ring H*(Gn; Z/2), pg. 83* 8. Existence of Stiefel-Whitney Classes, pg. 89* 9. Oriented Bundles and the Euler Class, pg. 95* 10. The Thom Isomorphism Theorem, pg. 105* 11. Computations in a Smooth Manifold, pg. 115* 12. Obstructions, pg. 139* 13. Complex Vector Bundles and Complex Manifolds, pg. 149* 14. Chern Classes, pg. 155* 15. Pontrjagin Classes, pg. 173* 16. Chern Numbers and Pontrjagin Numbers, pg. 183* 17. The Oriented Cobordism Ring OMEGA*, pg. 199* 18. Thom Spaces and Transversality, pg. 205* 19. Multiplicative Sequences and the Signature Theorem, pg. 219* 20. Combinatorial Pontrjagin Classes, pg. 231*Epilogue, pg. 249*Appendix A: Singular Homology and Cohomology, pg. 257*Appendix B: Bernoulli Numbers, pg. 281*Appendix C: Connections, Curvature, and Characteristic Classes, pg. 289*Bibliography, pg. 315*Index, pg. 325

Read more less

Book SynopsisAddresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies.Table of Contents*FrontMatter, pg. i*Contents, pg. v*Acknowledgments, pg. vii*1. Introduction, pg. 1*2. An Iterative Decomposition of Global Conformal Invariants: The First Step, pg. 19*3. The Second Step: The Fefferman-Graham Ambient Metric and the Nature of the Decomposition, pg. 71*4. A Result on the Structure of Local Riemannian Invariants: The Fundamental Proposition, pg. 135*5. The Inductive Step of the Fundamental Proposition: The Simpler Cases, pg. 211*6. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part I, pg. 297*7. The Inductive Step of the Fundamental Proposition: The Hard Cases, Part II, pg. 361*A. Appendix, pg. 403*Bibliography, pg. 443*Index of Authors and Terms, pg. 447*Index of Symbols, pg. 449

Read more less

Book SynopsisIn Hypo-Analytic Structures Franois Treves provides a systematic approach to the study of the differential structures on manifolds defined by systems of complex vector fields. Serving as his main examples are the elliptic complexes, among which the De Rham and Dolbeault are the best known, and the tangential Cauchy-Riemann operators. Basic geometriTable of ContentsPrefaceIFormally and Locally Integrable Structures. Basic Definitions3I.1Involutive systems of linear PDE defined by complex vector fields. Formally and locally integrable structures5I.2The characteristic set. Partial classification of formally integrable structures11I.3Strongly noncharacteristic, totally real, and maximally real submanifolds16I.4Noncharacteristic and totally characteristic submanifolds23I.5Local representations27I.6The associated differential complex32I.7Local representations in locally integrable structures39I.8The Levi form in a formally integrable structure46I.9The Levi form in a locally integrable structure49I.10Characteristics in real and in analytic structures56I.11Orbits and leaves. Involutive structures of finite type63I.12A model case: Tube structures68IILocal Approximation and Representation in Locally Integrable Structures73II.1The coarse local embedding76II.2The approximation formula81II.3Consequences and generalizations86II.4Analytic vectors94II.5Local structure of distribution solutions and of L-closed currents100II.6The approximate Poincare lemma104II.7Approximation and local structure of solutions based on the fine local embedding108II.8Unique continuation of solutions115IIIHypo-Analytic Structures. Hypocomplex Manifolds120III.1Hypo-analytic structures121III.2Properties of hypo-analytic functions128III.3Submanifolds compatible with the hypo-analytic structure130III.4Unique continuation of solutions in a hypo-analytic manifold137III.5Hypocomplex manifolds. Basic properties145III.6Two-dimensional hypocomplex manifolds152Appendix to Section III.6: Some lemmas about first-order differential operators159III.7A class of hypocomplex CR manifolds162IVIntegrable Formal Structures. Normal Forms167IV.1Integrable formal structures168IV.2Hormander numbers, multiplicities, weights. Normal forms174IV.3Lemmas about weights and vector fields178IV.4Existence of basic vector fields of weight - 1185IV.5Existence of normal forms. Pluriharmonic free normal forms. Rigid structures191IV.6Leading parts198VInvolutive Structures with Boundary201V.1Involutive structures with boundary202V.2The associated differential complex. The boundary complex209V.3Locally integrable structures with boundary. The Mayer-Vietoris sequence219V.4Approximation of classical solutions in locally integrable structures with boundary226V.5Distribution solutions in a manifold with totally characteristic boundary228V.6Distribution solutions in a manifold with noncharacteristic boundary235V.7Example: Domains in complex space246VILocal Integrability and Local Solvability in Elliptic Structures252VI.1The Bochner-Martinelli formulas253VI.2Homotopy formulas for [actual symbol not reproducible] in convex and bounded domains258VI.3Estimating the sup norms of the homotopy operators264VI.4Holder estimates for the homotopy operators in concentric balls269VI.5The Newlander-Nirenberg theorem281VI.6End of the proof of the Newlander-Nirenberg theorem287VI.7Local integrability and local solvability of elliptic structures. Levi flat structures291VI.8Partial local group structures297VI.9Involutive structures with transverse group action. Rigid structures. Tube structures303VIIExamples of Nonintegrability and of Nonsolvability312VII.1Mizohata structures314VII.2Nonsolvability and nonintegrability when the signature of the Levi form is |n - 2|319VII.3Mizohata structures on two-dimensional manifolds324VII.4Nonintegrability and nonsolvability when the cotangent structure bundle has rank one330VII.5Nonintegrability and nonsolvability in Lewy structures. The three-dimensional case337VII.6Nonintegrability in Lewy structures. The higher-dimensional case343VII.7Example of a CR structure that is not locally integrable but is locally integrable on one side348VIIINecessary Conditions for the Vanishing of the Cohomology. Local Solvability of a Single Vector Field352VIII.1Preliminary necessary conditions for exactness354VIII.2Exactness of top-degree forms358VIII.3A necessary condition for local exactness based on the Levi form364VIII.4A result about structures whose characteristic set has rank at most equal to one367VIII.5Proof of Theorem VIII.4.1373VIII.6Applications of Theorem VII

Read more less

Book SynopsisPresents an introduction to differential geometry through differential forms, emphasizing their applications in various areas of mathematics and physics. This work focuses on Stokes' theorem, the classical integral formulas and their applications to harmonic functions and topology.Table of ContentsElements of multilinear algebra Differential forms in ${\mathbb{R}}^n$ Vector analysis on manifolds Pfaffian systems Curves and surfaces in Euclidean 3-space Lie groups and homogeneous spaces Symplectic geometry and mechanics Elements of statistical mechanics and thermodynamics Elements of electrodynamics Bibliography Symbols Index.

Read more less

Book SynopsisThis book addresses an important class of mathematical problems (the Riemann problem) for first-order hyperbolic partial differential equations (PDEs), which arise when modeling wave propagation in applications such as fluid dynamics, traffic flow, acoustics, and elasticity.It covers the fundamental ideas related to classical Riemann solutions, including their special structure and the types of waves that arise, as well as the ideas behind fast approximate solvers for the Riemann problem.The emphasis is on the general ideas, but each chapter delves into a particular application. The book is available in electronic form as a collection of Jupyter notebooks that contain executable computer code and interactive figures and animations.

Read more less

Book SynopsisThis textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.Trade Review“The book … is intended ‘for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science.’ … The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.” (Jer-Chin Chuang, MAA Reviews, October 4, 2021)“The book is intended for incremental study and covers both basic concepts and more advanced ones. The former are thoroughly supported with theory and examples, and the latter are backed up with extensive reading lists and references. … Thanks to its design and approach style this is a timely and much needed addition that enables interdisciplinary bridges and the discovery of new applications for differential geometry.” (Corina Mohorian, zbMATH 1453.53001, 2021)Table of Contents1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups ⊛.- 6. The Structure of O(p,q) and SO(p, q).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data ⊛.- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps ⊛.- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula ⊛.- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.

Read more less

Book SynopsisThis volume originated in talks given in Cortona at the conference "Geometric aspects of harmonic analysis" held in honor of the 70th birthday of Fulvio Ricci. It presents timely syntheses of several major fields of mathematics as well as original research articles contributed by some of the finest mathematicians working in these areas. The subjects dealt with are topics of current interest in closely interrelated areas of Fourier analysis, singular integral operators, oscillatory integral operators, partial differential equations, multilinear harmonic analysis, and several complex variables.The work is addressed to researchers in the field.Table of Contents- An Extension Problem and Hardy Type Inequalities for the Grushin Operator. - Sharp Local Smoothing Estimates for Fourier Integral Operators. - On the Hardy–Littlewood Maximal Functions in High Dimensions: Continuous and Discrete Perspective. - Potential Spaces on Lie Groups. - On Fourier Restriction for Finite-Type Perturbations of the Hyperbolic Paraboloid. - On Young’s Convolution Inequality for Heisenberg Groups. - Young’s Inequality Sharpened. - Strongly Singular Integrals on Stratified Groups. - Singular Brascamp–Lieb: A Survey. - On the Restriction of Laplace–Beltrami Eigenfunctions and Cantor-Type Sets. - Basis Properties of the Haar System in Limiting Besov Spaces. - Obstacle Problems Generated by the Estimates of Square Function. - Of Commutators and Jacobians. - On Regularity and Irregularity of Certain Holomorphic Singular Integral Operators.

Read more less

Book SynopsisThis book consists of contributions from the participants of the Abel Symposium 2019 held in Ålesund, Norway. It was centered about applications of the ideas of symmetry and invariance, including equivalence and deformation theory of geometric structures, classification of differential invariants and invariant differential operators, integrability analysis of equations of mathematical physics, progress in parabolic geometry and mathematical aspects of general relativity.The chapters are written by leading international researchers, and consist of both survey and research articles. The book gives the reader an insight into the current research in differential geometry and Lie theory, as well as applications of these topics, in particular to general relativity and string theory.Table of ContentsFour-dimensional homogeneous generalizations of Einstein Metrics.- Conformal and isometric embeddings of gravitational instantons.- Recent results on closed G2-structures, by Anna Fino and Alberto Raffero.- Almost Zoll affine surfaces.- Distinguished curves and fist integrals on Poincare-Einstein and other conformally singular geometries.- A car as parabolic geometry.- Legendrian cone structures and contact prolongations.- The search for solitons on homogeneous spaces.- On Ricci negative Lie groups.- Semi-Riemannian cones.- Building new Einstein spaces by deforming symmetric Einstein spaces.- Remarks on highly supersymmetric backgrounds of 11-dimensional supergravity.- Krichever-Novikov type algebras.

Read more less

Book SynopsisThis book studies a class of monopoles defined by certain mild conditions, called periodic monopoles of generalized Cherkis–Kapustin (GCK) type. It presents a classification of the latter in terms of difference modules with parabolic structure, revealing a kind of Kobayashi–Hitchin correspondence between differential geometric objects and algebraic objects. It also clarifies the asymptotic behaviour of these monopoles around infinity.The theory of periodic monopoles of GCK type has applications to Yang–Mills theory in differential geometry and to the study of difference modules in dynamical algebraic geometry. A complete account of the theory is given, including major generalizations of results due to Charbonneau, Cherkis, Hurtubise, Kapustin, and others, and a new and original generalization of the nonabelian Hodge correspondence first studied by Corlette, Donaldson, Hitchin and Simpson.This work will be of interest to graduate students and researchers in differential and algebraic geometry, as well as in mathematical physics.Table of Contents. - Introduction. - Preliminaries. - Formal Difference Modules and Good Parabolic Structure. - Filtered Bundles. - Basic Examples of Monopoles Around Infinity. - Asymptotic Behaviour of Periodic Monopoles Around Infinity. - The Filtered Bundles Associated with Periodic Monopoles. - Global Periodic Monopoles of Rank One. - Global Periodic Monopoles and Filtered Difference Modules. - Asymptotic Harmonic Bundles and Asymptotic Doubly Periodic Instantons (Appendix).

Read more less

Book SynopsisThis text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material.Controllability and optimal control problems are considered for nonlinear nonholonomic systems on smooth manifolds, in particular, on Lie groups. For the controllability problem, the following questions are considered: controllability of linear systems, local controllability of nonlinear systems, Nagano–Sussmann Orbit theorem, Rashevskii–Chow theorem, Krener's theorem. For the optimal control problem, Filippov's theorem is stated, invariant formulation of Pontryagin maximum principle on manifolds is given, second-order optimality conditions are discussed, and the sub-Riemannian problem is studied in detail. Pontryagin maximum principle is proved for sub-Riemannian problems, solution to the sub-Riemannian problems on the Heisenberg group, the group of motions of the plane, and the Engel group is described.Table of Contents1. Introduction.- 2. Controllability problem.- 3. Optimal control problem.- 4. Solution to optimal control problems.- 5. Conclusion.- A. Elliptic integrals, functions and equation of pendulum.- Bibliography and further reading.- Index.

Read more less

Book Synopsis- Part I: Introduction to Poisson Geometry.- 1. A brief Introduction to Poisson Geometry.- Part II: Wonderful Varieties.- 2. Wonderful Varieties with a View Towards Poisson Geometry.- Part III: An Invitation to Singular Foliations.- 3. What is a singular foliation?.- 4. Canonical geometric and algebraic structures hidden behind a singular foliation.- 5. State of the Art and open questions.

Read more less

Book SynopsisChapter 1. Map-germs from the plane to the plane.- Chapter 2. Geometric deformations of discriminants.- Chapter 3. Geometric deformations of the fold and cusp.- Chapter 4. Ae-codimension 1 singularities.- Chapter 5. Ae-codimension 2 singularities.- Chapter 6. Apparent contours.- Chapter 7. Geometric invariants.

Read more less

Book SynopsisAs in the field of "Invariant Distances and Metrics in Complex Analysis" there was and is a continuous progress this is now the second extended edition of the corresponding monograph. This comprehensive book is about the study of invariant pseudodistances (non-negative functions on pairs of points) and pseudometrics (non-negative functions on the tangent bundle) in several complex variables. It is an overview over a highly active research area at the borderline between complex analysis, functional analysis and differential geometry. New chapters are covering the Wu, Bergman and several other metrics. The book considers only domains in Cn and assumes a basic knowledge of several complex variables. It is a valuable reference work for the expert but is also accessible to readers who are knowledgeable about several complex variables. Each chapter starts with a brief summary of its contents and continues with a short introduction. It ends with an "Exercises" and a "List of problems" section that gathers all the problems from the chapter. The authors have been highly successful in giving a rigorous but readable account of the main lines of development in this area.

Read more less

Book SynopsisThis book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry – as differential geometry in general – has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces. The second edition of this monograph leads the reader from introductory concepts to recent research. Since the publication of the first edition in 1993 there appeared important new contributions, like the solutions of two different affine Bernstein conjectures, due to Chern and Calabi, respectively. Moreover, a large subclass of hyperbolic affine spheres were classified in recent years, namely the locally strongly convex Blaschke hypersurfaces that have parallel cubic form with respect to the Levi-Civita connection of the Blaschke metric. The authors of this book present such results and new methods of proof.

Read more less

Book SynopsisThis is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert’s Problem IV. These collected works include Busemann’s most important published articles on these topics. Volume I of the collection features Busemann’s papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann’s papers on convexity and integral geometry, on Hilbert’s Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann’s work, documents from his correspondence and introductory essays written by leading specialists on Busemann’s work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry. Table of ContentsPreface.- Introduction to Volume I.- List of publications of Herbert Busemann.- Acknowledgements.- Essays.- A. Papadpoulos: Herbert Busemann (1905-1994).- A. Papadopoulos and M. Troyanov: On three early papers by Herbert Busemann on the foundations of geometry.- M. Troyanov: On Pasch's Axiom and Desargues' Theorem in Busemann's work.- V. N. Berestovskiy: Busemann's results, ideas, questions on locally compact homogeneous geodesic spaces.- A. Papadopoulos and S. Yamada: Busemann's problems on G-spaces.- Busemann's metric theory of timelike spaces.- A. Papadopoulos: Chronogeometry.- W. M. Boothby: Review of Busemann's book The geometry of Geodesics.- F. A. Ficken: Review of Busemann's book Metric Methods in Finsler Spaces and in the Foundations of Geometry.- Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces.

Read more less

Book SynopsisIn dem Buch wird die Kurven- und Flächentheorie im 3-dimensionalen euklidischen Raum behandelt und ein Maple-Programmpaket auf einer CD zum konkreten Arbeiten geliefert. Mit einer neuen Programmerweiterung und dem theoretischen Hintergrund unter www.mi.uni-koeln.de/mi/Forschung/Reckziegel/diffgeo_ext/index.html.Table of ContentsDer Raum der elementaren Differentialgeometrie - Maple-Arbeitsmethoden im IR - Ebenen-Kurventheorie - Räumliche Kurventheorie - Einführung in die Flächentheorie - Modellierung von Flächen und Riemannschen Gebieten mit Maple - Äußere Geometrie von Flächen - Innere Geometrie von Flächen - Eine kurze Einführung in Maple

Read more less

Book SynopsisEs gibt in der Differentialgeometrie von Kurven und FJachen zwei Betrachtungsweisen. Die eine, die man klassische Differentialgeometrie nennen konnte, entstand zusammen mit den Anfangen der Differential-und Integralrechnung. Grob gesagt studiert die klassische Differentialgeometrie lokale Eigenschaften von Kurven und FHichen. Dabei verstehen wir unter lokalen Eigenschaften solche, die nur vom Verhalten der Kurve oder Flache in der Umgebung eines Punktes abhiingen. Die Methoden, die sich als fUr das Studium solcher Eigenschaften geeignet erwiesen haben, sind die Methoden der Differentialrechnung. Aus diesem Grund sind die in der Differentialgeometrie untersuchten Kurven und Flachen durch Funktionen definiert, die von einer gewissen Differenzierbarkeitsklasse sind. Die andere Betrachtungsweise ist die sogenannte globale Differentialgeometrie. Hierbei untersucht man den EinfluB lokaler Eigenschaften auf das Verhalten der gesamten Kurve oder Flache. Der interessanteste und reprasentativste Teil der klassischen Differentialgeometrie ist wohl die Untersuchung von Flachen. Beim Studium von Flachen treten jedoch in nattirlicher Weise einige 10k ale Eigenschaften von Kurven auf. Deshalb benutzen wir dieses erste Kapi­ tel, urn kurz auf Kurven einzugehen.Table of Contents1 Kurven.- 1.1 Einleitung.- 1.2 Parametrisierte Kurven.- 1.3 Reguläre Kurven. Bogenlänge.- 1.4 Das Vektorprodukt in ?3.- 1.5 Die lokale Theorie von Kurven, die nach der Bogenlänge parametrisiert sind.- 1.6 Die lokale kanonische Form.- 1.7 Globale Eigenschaften ebener Kurven.- 2 Reguläre Flächen.- 2.1 Einleitung.- 2.2 Reguläre Flächen. Urbilder regulärer Werte.- 2.3 Parameterwechsel. Differenzierbare Funktionen auf Flächen.- 2.4 Die Tangentialebene. Das Differential einer Abbildung.- 2.5 Die erste Fundamentalform. Flächeninhalt.- 2.6 Orientierung von Flächen.- 2.7 Eine Charakterisierung kompakter orientierbarer Flächen.- 2.8 Eine geometrische Definition des Flächeninhalts.- 3 Die Geometrie der Gauß-Abbildung.- 3.1 Einleitung.- 3.2 Die Definition der Gauß-Abbildung und ihre fundamentalen Eigenschaften.- 3.3 Die Gauß-Abbildung in lokalen Koordinaten.- 3.4 Vektorfelder.- 3.5 Regelflächen und Minimalflächen.- 4 Die innere Geometrie von Flächen.- 4.1 Einleitung.- 4.2 Isometrie. Konforme Abbildungen.- 4.3 Der Satz von Gauß und die Verträglichkeitsbedingungen.- 4.4 Parallelverschiebung. Geodätische.- 4.5 Der Satz von Gauß-Bonnet und seine Anwendungen.- 4.6 Die Exponentialabbildung. Geodätische Polarkoordinaten.- 4.7 Weitere Eigenschaften von Geodätischen. Konvexe Umgebungen.- Anhang: Beweise der Fundamentalsätze der lokalen Kurven-und Flächentheorie.- Hinweise und Lösungen.- Kommentiertes Literaturverzeichnis.- Namen-und Sachwortverzeichnis.

Read more less

Book SynopsisThis book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.Table of Contents1 Riemannian Geometry.- 2 Sobolev Spaces.- 3 Background Material.- 4 Complementary Material.- 5 The Yamabe Problem.- 6 Prescribed Scalar Curvature.- 7 Einstein—Kähler Metrics.- 8 Monge—Ampère Equations.- 9 The Ricci Curvature.- 10 Harmonic Maps.- Bibliography*.- Notation.

Read more less

Book SynopsisFrom the preface: "Hopf algebras, Hopf fibration of spheres, Hopf-Rinow complete Riemannian manifolds, Hopf theorem on the ends of groups - can one imagine modern mathematics without all this? Many other concepts and methods, fundamental in various mathematical disciplines, also go back directly or indirectly to the work of Heinz Hopf: homological algebra, singularities of vector fields and characteristic classes, group-like spaces, global differential geometry, and the whole algebraisation of topology with its influence on group theory, analysis and algebraic geometry. It is astonishing to realize that this oeuvre of a whole scientific life consists of only about 70 writings. Astonishing also the transparent and clear style, the concreteness of the problems, and how abstract and far-reaching the methods Hopf invented."Trade Review Heinz Hopf (1894-1981) is rightly considered to be one of the outstanding and most influential mathematicians of the XXth century. He was a pioneer in algebraic topology as well as in differential geometry. He is widely known as having studied the ‘Hopf fibration’. The very general abstract notion of Hopf algebra was introduced as tracing in Hopf’s works; he may be considered to have been a forerunner of the creation of homological algebra. He found a noncontractible map of the 3-sphere into the 2-sphere; that result was an essential step towards the concept of ‘Hopf invariant’ and the popularization of the homotopy group notion due to Hurewicz. Heinz Hopf was born in Wroclaw (Breslau), in the then German part of Poland. He studied in his home town, in Heidelberg and in Berlin, visited Göttingen, Princeton University, and finally settled at ETH in Zürich, where he became Weyl’s successor. The Heinz Hopf Selecta published in 1964 contained an important – although far from being complete – part of Hopf’s mathematical production. So this volume presenting Hopf’s collected works is welcome. As one may expect, the organisational achievement by Beno Eckmann, Hopf’s student and friend, is high class. Two important articles are translated from German into English. This book of over 1200 pages featuring 71 items constitutes an essential reference for the development of mathematics during the XXth century. Jean-Paul Pier (Zbl. MATH 980, 01027)Table of ContentsTable of Contents.- List of Publications of Heinz Hopf.- Editor's Preface.- Papers of Heinz Hopf.- Heinz Hopf Selecta.

Read more less

Book SynopsisDieses Lehrbuch behandelt die zentralen Themen der Linearen Algebra einschließlich ihrer Anwendungen. Neben einer systematischen Einführung der Rechenoperationen mit Vektoren und Matrizen werden entsprechende Rechengesetze angegeben, und es wird erklärt, warum diese gelten. Zahlreiche sehr ausführlich vorgerechnete Beispiele machen das Lehrbuch zu einer wertvollen Basis für das Selbststudium oder zur Vorbereitung auf Prüfungen. Viele dieser Beispiele geben außerdem einen Einblick, welche Problemstellungen mittels der Vektor- und Matrizenrechnung behandelt werden können. Neben allgemeinen Lösungsstrategien für lineare Gleichungssysteme werden Lösungsalgorithmen diskutiert, welche auf spezifische Anwendungsgebiete abgestimmt sind – z. B. Algorithmen zur Lösung von tridiagonalen Gleichungssystemen, von Gleichungssystemen mit einer symmetrischen, positiv definiten Matrix und von Gleichungssystemen, die in der Ausgleichungsrechnung auftreten. Für eine ganze Reihe von Problemen wie der Lösung linearer Gleichungssysteme, der Berechnung von Determinanten und der Berechnung der Inversen einer Matrix werden verschiedene Algorithmen vorgestellt. Bei der Nutzung dieser unterschiedlichen Algorithmen zeigt sich, dass manche davon eine sehr hohe Rechenzeit erfordern, während man mit anderen das Rechenergebnis schon nach einer sehr geringen Rechenzeit erhält. Um einschätzen zu können, welche der Algorithmen wann bevorzugt eingesetzt werden sollten, wird für viele Algorithmen eine Analyse des Aufwandes an Rechenoperationen durchgeführt. Der Inhalt Vektoren – Matrizen – Rechnen mit Vektoren und Matrizen – allgemeine Lösungsalgorithmen für lineare Gleichungssysteme – Lösungsalgorithmen für spezielle Gleichungssysteme Die Zielgruppen Studierende der Natur- und IngenieurwissenschaftenTable of ContentsVektoren.- Matrizen und lineare Gleichungssysteme.- Index.- Literaturverzeichnis.

Read more less