Differentialgeometri og Riemannsk geometri

This monograph is based on the author's results on the Riemannian ge- ometry of foliations with nonnegative mixed curvature and on the geometry of sub manifolds with generators (rulings) in a Riemannian space of nonnegative curvature. The main idea is that such foliated (sub) manifolds can be decom- posed when the dimension of the leaves (generators) is large. The methods of investigation are mostly synthetic. The work is divided into two parts, consisting of seven chapters and three appendices. Appendix A was written jointly with V. Toponogov. Part 1 is devoted to the Riemannian geometry of foliations. In the first few sections of Chapter I we give a survey of the basic results on foliated smooth manifolds (Sections 1.1-1.3), and finish in Section 1.4 with a discussion of the key problem of this work: the role of Riemannian curvature in the study of foliations on manifolds and submanifolds.

Der vorliegende Klassiker bietet Studierenden und Forschenden in den Gebieten der Theoretischen und Mathematischen Physik eine ideale Einführung in die Differentialgeometrie und Topologie. Beides sind wichtige Werkzeuge in den Gebieten der Astrophysik, der Teilchen- und Festkörperphysik. Das Buch führt durch:- Pfadintegralmethode und Eichtheorie- Mathematische Grundlagen von Abbildungen, Vektorräumen und Topologie- Fortgeschrittene Konzepte der Geometrie und Topologie und deren Anwendungen im Bereich der Flüssigkristalle, bei suprafluidem Helium, in der ART und der bosonischen Stringtheorie- Eine Zusammenführung von Geometrie und Topologie: Faserbündel, charakteristische Klassen und Indextheoreme- Anwendungen von Geometrie und Topologie in der modernen Physik: Eichfeldtheorien und der Analyse der Polakov'schen bosonischen Stringtheorie aus einer geometrischen Perspektive

L'opera fornisce una introduzione alla geometria delle varietà differenziabili, illustrandone le principali proprietà e descrivendo le principali tecniche e i più importanti strumenti usati per il loro studio. Uno degli obiettivi primari dell'opera è di fungere da testo di riferimento per chi (matematici, fisici, ingegneri) usa la geometria differenziale come strumento; inoltre può essere usato come libro di testo per diversi corsi introduttivi alla geometria differenziale, concentrandosi su alcuni dei vari aspetti della teoria presentati nell'opera. Più in dettaglio, nell'opera saranno trattati i seguenti argomenti: richiami di algebra multilineare e tensoriale, spesso non presentati nei corsi standard di algebra lineare; varietà differenziali, incluso il teorema di Whitney; fibrati vettoriali, incluso il teorema di Frobenius e un'introduzione ai fibrati principali; gruppi di Lie, incluso il teorema di corrispondenza fra sottogruppi e sottoalgebre; coomologia di de Rham, inclusa la dualità di Poincaré e il teorema di de Rham; connessioni, inclusa la teoria delle geodetiche; e geometria Riemanniana, con particolare attenzione agli operatori di curvatura e inclusi teoremi di Cartan-Hadamard, Bonnet-Myers, e Synge-Weinstein. Come abitudine degli autori, il testo è scritto in modo da favorire una lettura attiva, cruciale per un buon apprendimento di argomenti matematici; inoltre è corredato da numerosi esempi svolti ed esercizi proposti.

During the last ten years a powerful technique for the study of partial differential equations with regular singularities has developed using the theory of hyperfunctions. The technique has had several important applications in harmonic analysis for symmetric spaces.This book gives an introductory exposition of the theory of hyperfunctions and regular singularities, and on this basis it treats two major applications to harmonic analysis. The first is to the proof of Helgason's conjecture, due to Kashiwara et al., which represents eigenfunctions on Riemannian symmetric spaces as Poisson integrals of their hyperfunction boundary values.A generalization of this result involving the full boundary of the space is also given. The second topic is the construction of discrete series for semisimple symmetric spaces, with an unpublished proof, due to Oshima, of a conjecture of Flensted-Jensen.This first English introduction to hyperfunctions brings readers to the forefront of research in the theory of harmonic analysis on symmetric spaces. A substantial bibliography is also included. This volume is based on a paper which was awarded the 1983 University of Copenhagen Gold Medal Prize.

From a historical point of view, the theory we submit to the present study has its origins in the famous dissertation of P. Finsler from 1918 ([Fi]). In a the classical notion also conventional classification, Finsler geometry has besides a number of generalizations, which use the same work technique and which can be considered self-geometries: Lagrange and Hamilton spaces. Finsler geometry had a period of incubation long enough, so that few math­ ematicians (E. Cartan, L. Berwald, S.S. Chem, H. Rund) had the patience to penetrate into a universe of tensors, which made them compare it to a jungle. To aU of us, who study nowadays Finsler geometry, it is obvious that the qualitative leap was made in the 1970's by the crystallization of the nonlinear connection notion (a notion which is almost as old as Finsler space, [SZ4]) and by work-skills into its adapted frame fields. The results obtained by M. Matsumoto (coUected later, in 1986, in a monograph, [Ma3]) aroused interest not only in Japan, but also in other countries such as Romania, Hungary, Canada and the USA, where schools of Finsler geometry are founded and are presently widely recognized.

In this book the authors develop and work out applications to gravity and gauge theories and their interactions with generic matter fields, including spinors in full detail. Spinor fields in particular appear to be the prototypes of truly gauge-natural objects, which are not purely gauge nor purely natural, so that they are a paradigmatic example of the intriguing relations between gauge natural geometry and physical phenomenology. In particular, the gauge natural framework for spinors is developed in this book in full detail, and it is shown to be fundamentally related to the interaction between fermions and dynamical tetrad gravity.

* Contains research and survey articles by well known and respected mathematicians on recent developments and research trends in differential geometry and topology* Dedicated in honor of Lieven Vanhecke, as a tribute to his many fruitful and inspiring contributions to these fields* Papers include all necessary introductory and contextual material to appeal to non-specialists, as well as researchers and differential geometers

Extrinsic geometry describes properties of foliations on Riemannian manifolds which can be expressed in terms of the second fundamental form of the leaves. The authors of Topics in Extrinsic Geometry of Codimension-One Foliations achieve a technical tour de force, which will lead to important geometric results.  The Integral Formulae, introduced in chapter 1, is a useful for problems such as: prescribing higher mean curvatures of foliations, minimizing volume and energy defined for vector or plane fields on manifolds, and existence of foliations whose leaves enjoy given geometric properties. The Integral Formulae steams from a Reeb formula, for foliations on space forms which generalize the classical ones. For a special auxiliary functions the formulae involve the Newton transformations of the Weingarten operator. The central topic of this book is Extrinsic Geometric Flow (EGF) on foliated manifolds, which may be a tool for prescribing extrinsic geometric properties of foliations. To develop EGF, one needs Variational Formulae, revealed in chapter 2, which expresses a change in different extrinsic geometric quantities of a fixed foliation under leaf-wise variation of the Riemannian Structure of the ambient manifold. Chapter 3 defines a general notion of EGF and studies the evolution of Riemannian metrics along the trajectories of this flow(e.g., describes the short-time existence and uniqueness theory and estimate the maximal existence time).Some special solutions (called Extrinsic Geometric Solutions) of EGF are presented and are of great interest, since they provide Riemannian Structures with very particular geometry of the leaves. This work is aimed at those who have an interest in the differential geometry of submanifolds and foliations of Riemannian manifolds.

This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions.The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors, transformations using matrices, and then studies more complex geometrical modeling problems related to analysis of curves and surfaces. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format.This text greatly extends the author's previous title, Geometry of Curves and Surfaces with Maple (Birkhäuser, (c) 2000), and has a different focus. In addition to being applications driven and motivated by numerous examples and exercises from real-world fields, the book also contains over 60 percent new material, including new sections with complex numbers, quaternions, matrices and transformations, hyperbolic geometry, fractals, and surface-splines and over 300 figures reproducible using MATLAB® programs.This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines, engineers, computer scientists, and instructors of appliedmathematics.

Stochastic Geometry is the mathematical discipline which studies mathematical models for random geometric structures, as they appear frequently in almost all natural sciences or technical fields. Although its roots can be traced back to the 18th century (the Buffon needle problem), the modern theory of random sets was founded by D. Kendall and G. Matheron in the early 1970's. Its rapid development was influenced by applications in Spatial Statistics and by its close connections to Integral Geometry. The volume "Stochastic Geometry" contains the lectures given at the CIME summer school in Martina Franca in September 1974. The four main lecturers covered the areas of Spatial Statistics, Random Points, Integral Geometry and Random Sets, they are complemented by two additional contributions on Random Mosaics and Crystallization Processes. The book presents an up-to-date description of important parts of Stochastic Geometry.