Global Analysis on Open Manifolds

Global Analysis on Open Manifolds
Author: Jürgen Eichhorn
Publisher: Nova Publishers
Total Pages: 664
Release: 2007
Genre: Mathematics
ISBN: 9781600215636

Global analysis is the analysis on manifolds. Since the middle of the sixties there exists a highly elaborated setting if the underlying manifold is compact, evidence of which can be found in index theory, spectral geometry, the theory of harmonic maps, many applications to mathematical physics on closed manifolds like gauge theory, Seiberg-Witten theory, etc. If the underlying manifold is open, i.e. non-compact and without boundary, then most of the foundations and of the great achievements fail. Elliptic operators are no longer Fredholm, the analytical and topological indexes are not defined, the spectrum of self-adjoint elliptic operators is no longer discrete, functional spaces strongly depend on the operators involved and the data from geometry, many embedding and module structure theorems do not hold, manifolds of maps are not defined, etc. It is the goal of this new book to provide serious foundations for global analysis on open manifolds, to discuss the difficulties and special features which come from the openness and to establish many results and applications on this basis.

The Convenient Setting of Global Analysis

The Convenient Setting of Global Analysis
Author: Andreas Kriegl
Publisher: American Mathematical Soc.
Total Pages: 631
Release: 1997
Genre: Mathematics
ISBN: 0821807803

For graduate students and research mathematicians interested in global analysis and the analysis of manifolds, lays the foundations for a differential calculus in infinite dimensions and discusses applications in infinite-dimension differential geometry and global analysis not involving Sobolev completions and fixed-point theory. Shows how the notion of smoothness as mapping smooth curves to smooth curves coincides with all known reasonable concepts up to Frechet spaces. Then develops a calculus of holomorphic mappings, and another of real analytical mapping. Emphasizes regular infinite dimensional Lie groups. Annotation copyrighted by Book News, Inc., Portland, OR

Handbook of Global Analysis

Handbook of Global Analysis
Author: Demeter Krupka
Publisher: Elsevier
Total Pages: 1243
Release: 2011-08-11
Genre: Mathematics
ISBN: 0080556736

This is a comprehensive exposition of topics covered by the American Mathematical Society’s classification “Global Analysis , dealing with modern developments in calculus expressed using abstract terminology. It will be invaluable for graduate students and researchers embarking on advanced studies in mathematics and mathematical physics.This book provides a comprehensive coverage of modern global analysis and geometrical mathematical physics, dealing with topics such as; structures on manifolds, pseudogroups, Lie groupoids, and global Finsler geometry; the topology of manifolds and differentiable mappings; differential equations (including ODEs, differential systems and distributions, and spectral theory); variational theory on manifolds, with applications to physics; function spaces on manifolds; jets, natural bundles and generalizations; and non-commutative geometry. - Comprehensive coverage of modern global analysis and geometrical mathematical physics- Written by world-experts in the field- Up-to-date contents

Advanced Real Analysis

Advanced Real Analysis
Author: Anthony W. Knapp
Publisher: Springer Science & Business Media
Total Pages: 484
Release: 2008-07-11
Genre: Mathematics
ISBN: 0817644423

* Presents a comprehensive treatment with a global view of the subject * Rich in examples, problems with hints, and solutions, the book makes a welcome addition to the library of every mathematician

Relative Index Theory, Determinants and Torsion for Open Manifolds

Relative Index Theory, Determinants and Torsion for Open Manifolds
Author: Jrgen Eichhorn
Publisher: World Scientific
Total Pages: 353
Release: 2009
Genre: Mathematics
ISBN: 981277145X

For closed manifolds, there is a highly elaborated theory of number-valued invariants, attached to the underlying manifold, structures and differential operators. On open manifolds, nearly all of this fails, with the exception of some special classes. The goal of this monograph is to establish for open manifolds, structures and differential operators an applicable theory of number-valued relative invariants. This is of great use in the theory of moduli spaces for nonlinear partial differential equations and mathematical physics. The book is self-contained: in particular, it contains an outline of the necessary tools from nonlinear Sobolev analysis.

The Convenient Setting of Global Analysis

The Convenient Setting of Global Analysis
Author: Andreas Kriegl
Publisher: American Mathematical Society
Total Pages: 631
Release: 2024-08-15
Genre: Mathematics
ISBN: 1470478935

This book lays the foundations of differential calculus in infinite dimensions and discusses those applications in infinite dimensional differential geometry and global analysis not involving Sobolev completions and fixed point theory. The approach is simple: a mapping is called smooth if it maps smooth curves to smooth curves. Up to Fr‚chet spaces, this notion of smoothness coincides with all known reasonable concepts. In the same spirit, calculus of holomorphic mappings (including Hartogs' theorem and holomorphic uniform boundedness theorems) and calculus of real analytic mappings are developed. Existence of smooth partitions of unity, the foundations of manifold theory in infinite dimensions, the relation between tangent vectors and derivations, and differential forms are discussed thoroughly. Special emphasis is given to the notion of regular infinite dimensional Lie groups. Many applications of this theory are included: manifolds of smooth mappings, groups of diffeomorphisms, geodesics on spaces of Riemannian metrics, direct limit manifolds, perturbation theory of operators, and differentiability questions of infinite dimensional representations.

Nonlinear Analysis on Manifolds. Monge-Ampère Equations

Nonlinear Analysis on Manifolds. Monge-Ampère Equations
Author: Thierry Aubin
Publisher: Springer Science & Business Media
Total Pages: 215
Release: 2012-12-06
Genre: Mathematics
ISBN: 1461257344

This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical studies are inadequate to solve them. Each problem has its own particular difficulties. Nevertheless there exist some standard methods which are useful and which we must know to apply them. One should not forget that our problems are motivated by geometry, and that a geometrical argument may simplify the problem under investigation. Examples of this kind are still too rare. This work is neither a systematic study of a mathematical field nor the presentation of a lot of theoretical knowledge. On the contrary, I do my best to limit the text to the essential knowledge. I define as few concepts as possible and give only basic theorems which are useful for our topic. But I hope that the reader will find this sufficient to solve other geometrical problems by analysis.

Analysis and Geometry on Graphs and Manifolds

Analysis and Geometry on Graphs and Manifolds
Author: Matthias Keller
Publisher: Cambridge University Press
Total Pages: 493
Release: 2020-08-20
Genre: Mathematics
ISBN: 1108587380

This book addresses the interplay between several rapidly expanding areas of mathematics. Suitable for graduate students as well as researchers, it provides surveys of topics linking geometry, spectral theory and stochastics.

Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds

Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds
Author: Taeyoung Lee
Publisher: Springer
Total Pages: 561
Release: 2017-08-14
Genre: Mathematics
ISBN: 3319569538

This book provides an accessible introduction to the variational formulation of Lagrangian and Hamiltonian mechanics, with a novel emphasis on global descriptions of the dynamics, which is a significant conceptual departure from more traditional approaches based on the use of local coordinates on the configuration manifold. In particular, we introduce a general methodology for obtaining globally valid equations of motion on configuration manifolds that are Lie groups, homogeneous spaces, and embedded manifolds, thereby avoiding the difficulties associated with coordinate singularities. The material is presented in an approachable fashion by considering concrete configuration manifolds of increasing complexity, which then motivates and naturally leads to the more general formulation that follows. Understanding of the material is enhanced by numerous in-depth examples throughout the book, culminating in non-trivial applications involving multi-body systems. This book is written for a general audience of mathematicians, engineers, and physicists with a basic knowledge of mechanics. Some basic background in differential geometry is helpful, but not essential, as the relevant concepts are introduced in the book, thereby making the material accessible to a broad audience, and suitable for either self-study or as the basis for a graduate course in applied mathematics, engineering, or physics.