The Geometry of the Group of Symplectic Diffeomorphism - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Geometry of the Group of Symplectic Diffeomorphism write by Leonid Polterovich. This book was released on 2012-12-06. The Geometry of the Group of Symplectic Diffeomorphism available in PDF, EPUB and Kindle. The group of Hamiltonian diffeomorphisms Ham(M, 0) of a symplectic mani fold (M, 0) plays a fundamental role both in geometry and classical mechanics. For a geometer, at least under some assumptions on the manifold M, this is just the connected component of the identity in the group of all symplectic diffeomorphisms. From the viewpoint of mechanics, Ham(M,O) is the group of all admissible motions. What is the minimal amount of energy required in order to generate a given Hamiltonian diffeomorphism I? An attempt to formalize and answer this natural question has led H. Hofer [HI] (1990) to a remarkable discovery. It turns out that the solution of this variational problem can be interpreted as a geometric quantity, namely as the distance between I and the identity transformation. Moreover this distance is associated to a canonical biinvariant metric on Ham(M, 0). Since Hofer's work this new ge ometry has been intensively studied in the framework of modern symplectic topology. In the present book I will describe some of these developments. Hofer's geometry enables us to study various notions and problems which come from the familiar finite dimensional geometry in the context of the group of Hamiltonian diffeomorphisms. They turn out to be very different from the usual circle of problems considered in symplectic topology and thus extend significantly our vision of the symplectic world.
Lectures on Symplectic Geometry
Lectures on Symplectic Geometry - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Lectures on Symplectic Geometry write by Ana Cannas da Silva. This book was released on 2004-10-27. Lectures on Symplectic Geometry available in PDF, EPUB and Kindle. The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.
The Structure of Classical Diffeomorphism Groups
The Structure of Classical Diffeomorphism Groups - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Structure of Classical Diffeomorphism Groups write by Augustin Banyaga. This book was released on 2013-03-14. The Structure of Classical Diffeomorphism Groups available in PDF, EPUB and Kindle. In the 60's, the work of Anderson, Chernavski, Kirby and Edwards showed that the group of homeomorphisms of a smooth manifold which are isotopic to the identity is a simple group. This led Smale to conjecture that the group Diff'" (M)o of cr diffeomorphisms, r ~ 1, of a smooth manifold M, with compact supports, and isotopic to the identity through compactly supported isotopies, is a simple group as well. In this monograph, we give a fairly detailed proof that DifF(M)o is a simple group. This theorem was proved by Herman in the case M is the torus rn in 1971, as a consequence of the Nash-Moser-Sergeraert implicit function theorem. Thurston showed in 1974 how Herman's result on rn implies the general theorem for any smooth manifold M. The key idea was to vision an isotopy in Diff'"(M) as a foliation on M x [0, 1]. In fact he discovered a deep connection between the local homology of the group of diffeomorphisms and the homology of the Haefliger classifying space for foliations. Thurston's paper [180] contains just a brief sketch of the proof. The details have been worked out by Mather [120], [124], [125], and the author [12]. This circle of ideas that we call the "Thurston tricks" is discussed in chapter 2. It explains how in certain groups of diffeomorphisms, perfectness leads to simplicity. In connection with these ideas, we discuss Epstein's theory [52], which we apply to contact diffeomorphisms in chapter 6.
Function Theory on Symplectic Manifolds
Function Theory on Symplectic Manifolds - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Function Theory on Symplectic Manifolds write by Leonid Polterovich. This book was released on 2014. Function Theory on Symplectic Manifolds available in PDF, EPUB and Kindle. This is a book on symplectic topology, a rapidly developing field of mathematics which originated as a geometric tool for problems of classical mechanics. Since the 1980s, powerful methods such as Gromov's pseudo-holomorphic curves and Morse-Floer theory on loop spaces gave rise to the discovery of unexpected symplectic phenomena. The present book focuses on function spaces associated with a symplectic manifold. A number of recent advances show that these spaces exhibit intriguing properties and structures, giving rise to an alternative intuition and new tools in symplectic topology. The book provides an essentially self-contained introduction into these developments along with applications to symplectic topology, algebra and geometry of symplectomorphism groups, Hamiltonian dynamics and quantum mechanics. It will appeal to researchers and students from the graduate level onwards.
The Geometry of the Group of Symplectic Diffeomorphisms
The Geometry of the Group of Symplectic Diffeomorphisms - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Geometry of the Group of Symplectic Diffeomorphisms write by Leonid Polterovich. This book was released on 2001. The Geometry of the Group of Symplectic Diffeomorphisms available in PDF, EPUB and Kindle. The group of symplectic diffeomorphisms of a symplectic manifold plays a fundamental role both in geometry and classical mechanics. What is the minimal amount of energy required in order to generate a given mechanical motion? This variational problem admits an interpretation in terms of a remarkable geometry on the group discovered by Hofer in 1990. Hofer's geometry serves as a source of interesting problems and gives rise to new methods and notions which extend significantly our vision of the symplectic world. In the past decade this new geometry has been intensively studied in the framework of symplectic topology with the use of modern techniques such as Gromov's theory of pseudo-holomorphic curves, Floer homology and Guillemin-Sternberg-Lerman theory of symplectic connections. Furthermore, it opens up the intriguing prospect of using an alternative geometric intuition in dynamics. The book provides an essentially self-contained introduction into these developments and includes recent results on diameter, geodesics and growth of one-parameter subgroups in Hofer's geometry, as well as applications to dynamics and ergodic theory. It is addressed to researchers and students from the graduate level onwards.
