Download or read book Topics in Extrinsic Geometry of Codimension One Foliations written by Vladimir Rovenski and published by Springer Science & Business Media. This book was released on 2011-07-26 with total page 129 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Topics in Extrinsic Geometry of Codimension One Foliations written by and published by . This book was released on 2011-08-07 with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Extrinsic Geometry of Foliations written by Vladimir Rovenski and published by Springer Nature. This book was released on 2021-05-22 with total page 319 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to geometric problems of foliation theory, in particular those related to extrinsic geometry, modern branch of Riemannian Geometry. The concept of mixed curvature is central to the discussion, and a version of the deep problem of the Ricci curvature for the case of mixed curvature of foliations is examined. The book is divided into five chapters that deal with integral and variation formulas and curvature and dynamics of foliations. Different approaches and methods (local and global, regular and singular) in solving the problems are described using integral and variation formulas, extrinsic geometric flows, generalizations of the Ricci and scalar curvatures, pseudo-Riemannian and metric-affine geometries, and 'computable' Finsler metrics. The book presents the state of the art in geometric and analytical theory of foliations as a continuation of the authors' life-long work in extrinsic geometry. It is designed for newcomers to the field as well as experienced geometers working in Riemannian geometry, foliation theory, differential topology, and a wide range of researchers in differential equations and their applications. It may also be a useful supplement to postgraduate level work and can inspire new interesting topics to explore.
Download or read book Geometry and its Applications written by Vladimir Rovenski and published by Springer. This book was released on 2014-05-05 with total page 247 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume has been divided into two parts: Geometry and Applications. The geometry portion of the book relates primarily to geometric flows, laminations, integral formulae, geometry of vector fields on Lie groups and osculation; the articles in the applications portion concern some particular problems of the theory of dynamical systems, including mathematical problems of liquid flows and a study of cycles for non-dynamical systems. This Work is based on the second international workshop entitled "Geometry and Symbolic Computations," held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as MapleTM and Mathematica® , as well as presentation of new results.
Download or read book Foliations 2012 Proceedings Of The International Conference written by Jesus A Alvarez Lopez and published by World Scientific. This book was released on 2013-10-25 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a compilation of new results and surveys on the current state of some aspects of the foliation theory presented during the conference “FOLIATIONS 2012”. It contains recent materials on foliation theory which is related to differential geometry, the theory of dynamical systems and differential topology. Both the original research and survey articles found in here should inspire students and researchers interested in foliation theory and the related fields to plan his/her further research.
Download or read book Introduction to the Geometry of Foliations Part B written by Gilbert Hector and published by Vieweg+Teubner Verlag. This book was released on 1983-01-01 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Some Topics on Geometric Theory of Foliations written by Bruno Azevedo Scuardua and published by . This book was released on 2002 with total page 64 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Geometry of Foliations written by Philippe Tondeur and published by Birkhäuser. This book was released on 2012-12-06 with total page 308 pages. Available in PDF, EPUB and Kindle. Book excerpt: The topics in this survey volume concern research done on the differential geom etry of foliations over the last few years. After a discussion of the basic concepts in the theory of foliations in the first four chapters, the subject is narrowed down to Riemannian foliations on closed manifolds beginning with Chapter 5. Following the discussion of the special case of flows in Chapter 6, Chapters 7 and 8 are de voted to Hodge theory for the transversal Laplacian and applications of the heat equation method to Riemannian foliations. Chapter 9 on Lie foliations is a prepa ration for the statement of Molino's Structure Theorem for Riemannian foliations in Chapter 10. Some aspects of the spectral theory for Riemannian foliations are discussed in Chapter 11. Connes' point of view of foliations as examples of non commutative spaces is briefly described in Chapter 12. Chapter 13 applies ideas of Riemannian foliation theory to an infinite-dimensional context. Aside from the list of references on Riemannian foliations (items on this list are referred to in the text by [ ]), we have included several appendices as follows. Appendix A is a list of books and surveys on particular aspects of foliations. Appendix B is a list of proceedings of conferences and symposia devoted partially or entirely to foliations. Appendix C is a bibliography on foliations, which attempts to be a reasonably complete list of papers and preprints on the subject of foliations up to 1995, and contains approximately 2500 titles.
Download or read book Mathematical Reviews written by and published by . This book was released on 2007 with total page 840 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Foliations and the Geometry of 3 Manifolds written by Danny Calegari and published by Oxford University Press on Demand. This book was released on 2007-05-17 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: This unique reference, aimed at research topologists, gives an exposition of the 'pseudo-Anosov' theory of foliations of 3-manifolds. This theory generalizes Thurston's theory of surface automorphisms and reveals an intimate connection between dynamics, geometry and topology in 3 dimensions. Significant themes returned to throughout the text include the importance of geometry, especially the hyperbolic geometry of surfaces, the importance of monotonicity, especially in1-dimensional and co-dimensional dynamics, and combinatorial approximation, using finite combinatorical objects such as train-tracks, branched surfaces and hierarchies to carry more complicated continuous objects.
Download or read book Introduction to Differential Geometry written by Joel W. Robbin and published by Springer Nature. This book was released on 2022-01-12 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.
Download or read book Noncommutative Geometry written by Alain Connes and published by Springer. This book was released on 2003-12-15 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Noncommutative Geometry is one of the most deep and vital research subjects of present-day Mathematics. Its development, mainly due to Alain Connes, is providing an increasing number of applications and deeper insights for instance in Foliations, K-Theory, Index Theory, Number Theory but also in Quantum Physics of elementary particles. The purpose of the Summer School in Martina Franca was to offer a fresh invitation to the subject and closely related topics; the contributions in this volume include the four main lectures, cover advanced developments and are delivered by prominent specialists.
Download or read book Problems on Mapping Class Groups and Related Topics written by Benson Farb and published by American Mathematical Soc.. This book was released on 2006-09-12 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.
Download or read book The Problem of Time written by Edward Anderson and published by Springer. This book was released on 2017-09-18 with total page 917 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is a treatise on time and on background independence in physics. It first considers how time is conceived of in each accepted paradigm of physics: Newtonian, special relativity, quantum mechanics (QM) and general relativity (GR). Substantial differences are moreover uncovered between what is meant by time in QM and in GR. These differences jointly source the Problem of Time: Nine interlinked facets which arise upon attempting concurrent treatment of the QM and GR paradigms, as is required in particular for a background independent theory of quantum gravity. A sizeable proportion of current quantum gravity programs - e.g. geometrodynamical and loop quantum gravity approaches to quantum GR, quantum cosmology, supergravity and M-theory - are background independent in this sense. This book's foundational topic is thus furthermore of practical relevance in the ongoing development of quantum gravity programs. This book shows moreover that eight of the nine facets of the Problem of Time already occur upon entertaining background independence in classical (rather than quantum) physics. By this development, and interpreting shape theory as modelling background independence, this book further establishes background independence as a field of study. Background independent mechanics, as well as minisuperspace (spatially homogeneous) models of GR and perturbations thereabout are used to illustrate these points. As hitherto formulated, the different facets of the Problem of Time greatly interfere with each others' attempted resolutions. This book explains how, none the less, a local resolution of the Problem of Time can be arrived at after various reconceptualizations of the facets and reformulations of their mathematical implementation. Self-contained appendices on mathematical methods for basic and foundational quantum gravity are included. Finally, this book outlines how supergravity is refreshingly different from GR as a realization of background independence, and what background independence entails at the topological level and beyond.
Download or read book The Geometry and Topology of Three Manifolds written by William P. Thurston and published by American Mathematical Society. This book was released on 2023-06-16 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: William Thurston's work has had a profound influence on mathematics. He connected whole mathematical subjects in entirely new ways and changed the way mathematicians think about geometry, topology, foliations, group theory, dynamical systems, and the way these areas interact. His emphasis on understanding and imagination in mathematical learning and thinking are integral elements of his distinctive legacy. This four-part collection brings together in one place Thurston's major writings, many of which are appearing in publication for the first time. Volumes I–III contain commentaries by the Editors. Volume IV includes a preface by Steven P. Kerckhoff. Volume IV contains Thurston's highly influential, though previously unpublished, 1977–78 Princeton Course Notes on the Geometry and Topology of 3-manifolds. It is an indispensable part of the Thurston collection but can also be used on its own as a textbook or for self-study.
Download or read book Cartan for Beginners written by Thomas Andrew Ivey and published by American Mathematical Soc.. This book was released on 2003 with total page 394 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior differential systems and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems. It begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics.One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. The book features an introduction to $G$-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs via Darboux's method, the method of characteristics, and Cartan's method of equivalence. This text is suitable for a one-year graduate course in differential geometry, and parts of it can be used for a one-semester course. It has numerous exercises and examples throughout. It will also be useful to experts in areas such as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.
Download or read book Differential Topology written by Victor Guillemin and published by American Mathematical Soc.. This book was released on 2010 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential Topology provides an elementary and intuitive introduction to the study of smooth manifolds. In the years since its first publication, Guillemin and Pollack's book has become a standard text on the subject. It is a jewel of mathematical exposition, judiciously picking exactly the right mixture of detail and generality to display the richness within. The text is mostly self-contained, requiring only undergraduate analysis and linear algebra. By relying on a unifying idea--transversality--the authors are able to avoid the use of big machinery or ad hoc techniques to establish the main results. In this way, they present intelligent treatments of important theorems, such as the Lefschetz fixed-point theorem, the Poincaré-Hopf index theorem, and Stokes theorem. The book has a wealth of exercises of various types. Some are routine explorations of the main material. In others, the students are guided step-by-step through proofs of fundamental results, such as the Jordan-Brouwer separation theorem. An exercise section in Chapter 4 leads the student through a construction of de Rham cohomology and a proof of its homotopy invariance. The book is suitable for either an introductory graduate course or an advanced undergraduate course.
