Foliations: Dynamics, Geometry and Topology

ISBN-13: 9783034808705 / Angielski / Miękka / 2014 / 198 str.

Masayuki Asaoka

Opis

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.

Recenzje

“This book contains the lecture notes of four courses on several topics with rather different flavor, which are linked by their relation with Foliation Theory. … the courses will be very helpful for any reader that wants to get quickly introduced to any of these lines of research.” (Jesus A. Álvarez López, zbMATH 1318.57001, 2015)

Spis treści

Fundamentals of Foliation Theory.- Foliation Dynamics.- Deformation of Locally Free Actions and Leafwise Cohomology.- Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds.​

Blurb

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 arising and used in the study of foliations.

The lectures by A. El Kacimi Alaoui offer an introduction to Foliation Theory, with emphasis on examples and transverse structures. S. Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations, like limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, stable manifolds, Pesin Theory, and hyperbolic, parabolic, and elliptic types of foliations, all of them illustrated with examples. The lectures by M. Asaoka are devoted to the computation of the leafwise cohomology of orbit foliations given by locally free actions of certain Lie groups, and its application to the description of the deformation of those actions. In the lectures by K. Richardson, he studies the geometric and analytic 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 appeal to mathematicians interested in the applications to foliations of subjects like topology of manifolds, dynamics, cohomology or global analysis.