Bei der Herausgabe dieses Buches mochte ich an dieser Stelle Herrn L. Berwald in Prag, Herrn D. J. Struik in Delft und Herrn R. WeitzenbOck in Blaricum, die mich durch das Mitlesen der Korrek turen sowie durch viele wichtige Bemerkungen aufs wirksamste unter stiitzt haben, meinen verbindlichsten Dank aussprechen. Einen freundschaftlichen GruB dem mathematischen Kreise in Hamburg, wo es mir vergonnt war, im Sommersemester dieses Jahres iiber die mehrdimensionale Affingeometrie zu lesen. Manche anregende Bemerkung zum vierten Abschnitt brachte mir diese schOne Zeit, die mir immer in freudiger Erinnerung bleiben wird. Der Verlagsbuchhandlung Julius Springer meinen besonderen Dank fiir die sorgfaltige Behandlung der Korrekturen, die mir die sauere Arbeit des Korrigierens fast zu einer Freude machte. Delft, im Dezember 1923. J. A. Schouten. Inhaltsverzeichnis. Seite Einleitung . . . 1 I. Der algebraische Tei des Kalkiils. 1. Die allgemeine Mannigfaltigkeit Xn . . 8 2. Der Begriff der Ubertragung . . . . . . 9 3. Die euklidischaffine Mannigfaltigkeit En . 9 4. Kontravariante und kovariante Vektoren . 12 5. Kontravariante und kovariante Bivektoren, Trivektoren usw. 17 6. Geometrische Darstellung kontravarianter und kovarianter p-Vektoren bei Einschrankung der Gruppe 20 7. Allgemeine GriiBen . . . . . . . . . . 23 8. Die Uberschiebungen . . . . . . . . . 28 9. Geometrische Darstellung der Tensoren 32 10. GriiBen zweiten Grades und lineare Transformationen 33 11. Die Einfiihrung einer MaBbestimmung in der En . . 36 12. Die Fundamentaltensoren. . . . . . . . . . . . . 38 13. Geometrische Darstellung alternierender GriiBen bei der orthogonalen und rotationalen Gruppe. Metrische Eigenschaften . . . . . . . . 41 14. Metrische Eigenschaften eines Te-nsors zweiten Grades. . . . . . ."

An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. 99 illustrations.

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings:

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard's theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures. Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Study 79 contains a collection of papers presented at the Conference on Discontinuous Groups and Ricmann Surfaces at the University of Maryland, May 21-25, 1973. The papers, by leading authorities, deal mainly with Fuchsian and Kleinian groups, Teichmuller spaces, Jacobian varieties, and quasiconformal mappings. These topics are intertwined, representing a common meeting of algebra, geometry, and analysis.

'In a class populated by students who already have some exposure to the concept of a manifold, the presence of chapter 3 in this text may make for an unusual and interesting course. The primary function of this book will be as a text for a more conventional course in the classical theory of curves and surfaces.'MAA ReviewsThis engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well.Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates.Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities.In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field.

Over the last number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. A significantly revised second edition was published in 1998 introducing new sections and updates on the fast-developing area. This new third edition includes updates and new material to bring the book right up-to-date.

General relativity came to life in 1915, when Albert Einstein formulated his field equations. These unify space, time, and gravitation, where the latter acts through curvature. Thereby the laws of physics obtain a geometric nature. Mathematical general relativity investigates spacetimes, which are manifolds equipped with a Lorentzian metric obeying the related Einstein-matter systems of nonlinear partial differential equations. The fruitful interactions of mathematics and physics in general relativity have produced breakthroughs in all the related research fields. This volume includes 14 articles presenting aspects of the most important general relativity research of the past 100 years. Among them we find cosmological and non-cosmological questions, the Cauchy problem for the Einstein equations, stability results, black holes and their formation, gravitational waves and their memory effect, the concept of energy, and the asymptotics of spacetimes. Through geometric analysis and advanced theory of nonlinear partial differential equations, long-standing problems have been solved lately that had remained locked since the beginnings of this beautiful theory of general relativity. Many more burning questions have been formulated and are yet to be answered.