Foliations On Riemannian Manifolds

Author : Philippe Tondeur
Publisher : Springer Science & Business Media
Page : 258 pages
File Size : 53,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461387800

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Foliations on Riemannian Manifolds by Philippe Tondeur Pdf

A first approximation to the idea of a foliation is a dynamical system, and the resulting decomposition of a domain by its trajectories. This is an idea that dates back to the beginning of the theory of differential equations, i.e. the seventeenth century. Towards the end of the nineteenth century, Poincare developed methods for the study of global, qualitative properties of solutions of dynamical systems in situations where explicit solution methods had failed: He discovered that the study of the geometry of the space of trajectories of a dynamical system reveals complex phenomena. He emphasized the qualitative nature of these phenomena, thereby giving strong impetus to topological methods. A second approximation is the idea of a foliation as a decomposition of a manifold into submanifolds, all being of the same dimension. Here the presence of singular submanifolds, corresponding to the singularities in the case of a dynamical system, is excluded. This is the case we treat in this text, but it is by no means a comprehensive analysis. On the contrary, many situations in mathematical physics most definitely require singular foliations for a proper modeling. The global study of foliations in the spirit of Poincare was begun only in the 1940's, by Ehresmann and Reeb.

Author : Vladimir Rovenski
Publisher : Springer Science & Business Media
Page : 296 pages
File Size : 54,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461242703

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Foliations on Riemannian Manifolds and Submanifolds by Vladimir Rovenski Pdf

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.

Author : Molino
Publisher : Springer Science & Business Media
Page : 348 pages
File Size : 48,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781468486704

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Riemannian Foliations by Molino Pdf

Foliation theory has its origins in the global analysis of solutions of ordinary differential equations: on an n-dimensional manifold M, an [autonomous] differential equation is defined by a vector field X ; if this vector field has no singularities, then its trajectories form a par tition of M into curves, i.e. a foliation of codimension n - 1. More generally, a foliation F of codimension q on M corresponds to a partition of M into immersed submanifolds [the leaves] of dimension ,--------,- - . - -- p = n - q. The first global image that comes to mind is 1--------;- - - - - - that of a stack of "plaques". 1---------;- - - - - - Viewed laterally [transver 1--------1- - - -- sally], the leaves of such a 1--------1 - - - - -. stacking are the points of a 1--------1--- ----. quotient manifold W of di L..... -' _ mension q. -----~) W M Actually, this image corresponds to an elementary type of folia tion, that one says is "simple". For an arbitrary foliation, it is only l- u L ally [on a "simpIe" open set U] that the foliation appears as a stack of plaques and admits a local quotient manifold. Globally, a leaf L may - - return and cut a simple open set U in several plaques, sometimes even an infinite number of plaques.

Author : Enrique Macias-virgos,Jesus A Alvarez Lopez,Xose Masa
Publisher : World Scientific
Page : 258 pages
File Size : 42,8 Mb
Release : 1995-11-17
Category : Electronic
ISBN : 9789814549615

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Analysis And Geometry In Foliated Manifolds - Proceedings Of The 7th International Colloquium On Differential Geometry by Enrique Macias-virgos,Jesus A Alvarez Lopez,Xose Masa Pdf

The subject of this volume, recent developments in foliation theory and important related analytic and geometric techniques, is an active field in the application of both global analysis and geometric topological theory of manifolds to the study of foliations. This volume includes research papers by leading specialists, giving an overview of this subject.

Author : Paweł Walczak,Lawrence Conlon,Rémi Langevin,Takashi Tsuboi
Publisher : World Scientific
Page : 460 pages
File Size : 48,6 Mb
Release : 2002-02-01
Category : Electronic
ISBN : 9789814489706

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Foliations: Geometry and Dynamics by Paweł Walczak,Lawrence Conlon,Rémi Langevin,Takashi Tsuboi Pdf

This volume contains surveys and research articles regarding different aspects of the theory of foliation. The main aspects concern the topology of foliations of low-dimensional manifolds, the geometry of foliated Riemannian manifolds and the dynamical properties of foliations. Among the surveys are lecture notes devoted to the analysis of some operator algebras on foliated manifolds and the theory of confoliations (objects defined recently by W Thurston and Y Eliashberg, situated between foliations and contact structures). Among the research articles one can find a detailed proof of an unpublished theorem (due to Duminy) concerning ends of leaves in exceptional minimal sets. Contents:Survey Articles:Some Results on Secondary Characteristic Classes of Transversely Holomorphic Foliations (T Asuke)LS-Categories for Foliated Manifolds (H Colman)Dynamics and the Godbillon–Vey Class: A History and Survey (S Hurder)Similarity and Conformal Geometry of Foliations (R Langevin)Foliations and Contact Structures on 3-Manifolds (Y Mitsumatsu)Operator Algebras and the Index Theorem on Foliated Manifolds (H Moriyoshi)Research Articles:Distributional Betti Numbers of Transitive Foliations of Codimension One (J Álvarez-López & Y Kordyukov)Tautly Foliated 3-Manifolds with No R-Covered Foliations (M Brittenham)Endests of Exceptional Leaves — A Theorem of G Duminy (J Cantwell & L Conlon)Foliations and Compactly Generated Pseudogroups (A Haefliger)Transverse Lusternik–Schnirelmann Category and Non-Proper Leaves (R Langevin & P Walczak)On Exact Poisson Manifolds of Dimension 3 (T Mizutani)On the Perfectness of Groups of Diffeomorphisms of the Interval Tangent to the Identity at the Endpoints (T Tsuboi)and other papers Readership: Researchers interested in mathematics, especially in fields related to differential geometry and topology, and the theory of dynamical systems. Keywords:Proceedings;Workshop;Geometry;Warsaw (Poland);Dynamics;Euroworkshop

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Science & Business Media
Page : 129 pages
File Size : 44,6 Mb
Release : 2011-07-26
Category : Mathematics
ISBN : 9781441999085

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Topics in Extrinsic Geometry of Codimension-One Foliations by Vladimir Rovenski,Paweł Walczak Pdf

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.

Author : Pawel Walczak,Remi Langevin,Takashi Tsuboi,Steven Hurder
Publisher : World Scientific
Page : 490 pages
File Size : 40,6 Mb
Release : 2006-09-20
Category : Mathematics
ISBN : 9789814476782

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Foliations 2005 - Proceedings Of The International Conference by Pawel Walczak,Remi Langevin,Takashi Tsuboi,Steven Hurder Pdf

This volume takes a look at the current state of the theory of foliations, with surveys and research articles concerning different aspects. The focused aspects cover geometry of foliated Riemannian manifolds, Riemannian foliations and dynamical properties of foliations and some aspects of classical dynamics related to the field. Among the articles readers may find a study of foliations which admit a transverse contractive flow, an extensive survey on non-commutative geometry of Riemannian foliations, an article on contact structures converging to foliations, as well as a few articles on conformal geometry of foliations. This volume also contains a list of open problems in foliation theory which were collected from the participants of the Foliations 2005 conference.

Author : Vladimir Rovenski,Paweł Walczak
Publisher : Springer Nature
Page : 319 pages
File Size : 43,9 Mb
Release : 2021-05-22
Category : Mathematics
ISBN : 9783030700676

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Extrinsic Geometry of Foliations by Vladimir Rovenski,Paweł Walczak Pdf

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.

Author : Tadayoshi Mizutani,Kazuo Masuda,Shigenori Matsumoto,Takashi Inaba,Takashi Tsuboi,Yoshihiko Mitsumatsu
Publisher : World Scientific
Page : 514 pages
File Size : 46,6 Mb
Release : 1994-12-16
Category : Electronic
ISBN : 9789814550390

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Geometric Study Of Foliations - Proceedings Of The International Symposium/workshop by Tadayoshi Mizutani,Kazuo Masuda,Shigenori Matsumoto,Takashi Inaba,Takashi Tsuboi,Yoshihiko Mitsumatsu Pdf

This book covers recent topics in various aspects of foliation theory and its relation with other areas including dynamical systems, C∗-algebras, index theory and low-dimensional topology. It contains survey articles by G Hector, S Hurder and P Molino, as well as more than 20 original papers by specialists who are currently most active in the field.

Author : Philippe Tondeur
Publisher : Birkhäuser
Page : 308 pages
File Size : 40,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034889148

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Geometry of Foliations by Philippe Tondeur Pdf

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.

Author : Aurel Bejancu,Hani Reda Farran
Publisher : Springer Science & Business Media
Page : 309 pages
File Size : 48,8 Mb
Release : 2006-01-17
Category : Mathematics
ISBN : 9781402037207

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Foliations and Geometric Structures by Aurel Bejancu,Hani Reda Farran Pdf

Offers basic material on distributions and foliations. This book introduces and builds the tools needed for studying the geometry of foliated manifolds. Its main theme is to investigate the interrelations between foliations of a manifold on the one hand, and the many geometric structures that the manifold may admit on the other hand.

Author : Gilbert Hector
Publisher : Springer-Verlag
Page : 309 pages
File Size : 46,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9783322856197

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Introduction to the Geometry of Foliations, Part B by Gilbert Hector Pdf

Author : Masayuki Asaoka,Aziz El Kacimi Alaoui,Steven Hurder,Ken Richardson
Publisher : Springer
Page : 198 pages
File Size : 49,9 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9783034808712

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Foliations: Dynamics, Geometry and Topology by Masayuki Asaoka,Aziz El Kacimi Alaoui,Steven Hurder,Ken Richardson Pdf

This book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods converging in the study of foliations. The lectures by Aziz El Kacimi Alaoui provide an introduction to Foliation Theory with emphasis on examples and transverse structures. Steven Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations: limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, Pesin Theory and hyperbolic, parabolic and elliptic types of foliations. The lectures by Masayuki Asaoka compute the leafwise cohomology of foliations given by actions of Lie groups, and apply it to describe deformation of those actions. In his lectures, Ken Richardson studies the properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formula. Besides students and researchers of Foliation Theory, this book will be interesting for mathematicians interested in the applications to foliations of subjects like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or Global Analysis.

Author : Jesus A Alvarez Lopez,Steven Hurder,Remi Langevin,Takashi Tsuboi,Pawel Walczak
Publisher : World Scientific
Page : 276 pages
File Size : 48,6 Mb
Release : 2013-10-25
Category : Mathematics
ISBN : 9789814556873

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Foliations 2012 - Proceedings Of The International Conference by Jesus A Alvarez Lopez,Steven Hurder,Remi Langevin,Takashi Tsuboi,Pawel Walczak Pdf

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.

Author : Detlef Gromoll,Gerard Walschap
Publisher : Springer Science & Business Media
Page : 185 pages
File Size : 46,7 Mb
Release : 2009-03-28
Category : Mathematics
ISBN : 9783764387150

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Metric Foliations and Curvature by Detlef Gromoll,Gerard Walschap Pdf

Riemannian manifolds, particularly those with positive or nonnegative curvature, are constructed from only a handful by means of metric fibrations or deformations thereof. This text documents some of these constructions, many of which have only appeared in journal form. The emphasis is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.