Some problems of global analysis on asymptotically simple manifolds (original) (raw)
Related papers
Notes on Differentiable Manifolds
HARDWARI PUBLICATIONS, ALLAHABAD/PRAYAGRAJ (INDIA), 2023
PREFACE Padmashri Prof. Dr. R.S. Mishra (the Ph.D. supervisor of RBM and the great grand teacher of Ram Niwas) is credited to introduce researches in Differential Geometry as early as in 1947. Staring from then prevailing topics of classical 3-dimesional Differential Geometry such as Congru- ences etc., he developed his intellect to a very high degree and worked immensely in four diverse fields: Differential Geometry, Theory of Relativity, Theory of Shock Waves and Differential Manifolds. Some of his works (Texts, Research Monographs and the research papers) dwelling upon the topics pertain to the contents of this text are listed in the Bibliography. The mathematical community at large shall ever remain indebted to him to have created such a vast literature that the generations will be able to benefit themselves. It was a divine will that the first author, though a top ranker of Lucknow University, having failed in getting his research supervisor at Lucknow University was forced to go to Prof. Mishra at University of Allahabad, who was completely stranger to him. At this advanced age (81) he finds some truth in his father’s belief for being a gifted child of Rishi Bhāraḑwāj of Prayagraj. So, everything was under control of the divine forces to have united him to a legendary mathema- tician: Prof. Mishra, who was as versatile as the ‘Lord Shiva’ for being named (as Shankar – a synonym of Shiva) after the deity. The first author ever wished and made sincerest efforts to inculcate some of the merits of his Guru who still remains a fairy legend. Prof. Mishra had taken a great leap forward, which ever remains unfathomable by any of his students. RBM always wished to contribute something in this newer domain but, because of his limitations, his wish ever remained unfulfilled. Perhaps it is again a divine will that his alma mater recalled him after a long time (of six decades) to evaluate the younger inquisitive minds preparing for their Master’s examinations. Thus, sharpening his talent, RBM read the subject of his own rigorously and presented these Notes for the benefit of younger minds. The discipline being complex and most challenging of highest nature has been made as lucid as possible. Certain concepts such as Charts, Atlas, Projections, Tangent Surface, Vector Bundles, Contact Manifolds, etc. ever hunt the minds of explorers. The authors feel contended to have humbly presented the topics in the manner easy to comprehend. The subject being of advanced level its study requires the knowledge of Algebra, Linear Algebra, Differential Geometry, Topology and alike. Thus, the book comprises of two parts: The first part (having 10 Chapters) includes a brief discussion of pre-requites such as: (i) Number System; (ii) Plain (Euclidean) geometry, (iii) Matrices, (iv) Algebraic Structures (Sets and Functions, Groups, Rings, Fields, Integral domains); (v) Linear (or Vector) spaces, (vi) Metric spaces, (vii) Topological spaces, and (viii) Linear Algebra. Many topics in this Part are dealt with details such as Complex numbers, Algebraic Structures, Metric spaces, Topological spaces, etc., while the rest present basic concepts, important results (need not include the proofs). Part 2 consists of the five chapters: of which the Chapter 11 deals with the theory of manifolds. It covers the basic concepts, type of manifolds, their various aspects: topological, symplectic, differentiability etc. are covered. Kähler manifolds are introduced in the Chapter 12 which also includes a discussion of Sasakian manifolds. Theory of ‘Tangent Bundles and Vector Bundles’ is presented in Chapter 13. It also includes the Tensor Bundles. Contact Manifolds are presented in the next chapter while the Tachibana and Otsuki spaces form the subject matter of the last chapter. The course contents may best suit graduate and postgraduate programmes of any University and can be covered in one semester with 3 credit hours per week. These topics dealt in the first part are also included in the syllabi of various competitive examinations held in the Indian subcontinent. Chapters are arranged into Sections numbered chapter wise. The discussion within the Section is presented in the form of Definitions, Theorems, Notes and Examples. These subtitles within the Sections are numbered in decimal pattern. For instance, the equation number (C.S.E.) refers to the E th equation in the S th Section of Chapter C. When the number C coincides with the chapter at hand, it is not mentioned while quoting elsewhere. Foot-notes are arranged serially and are explained at the end of the chapter concerned. Normally, the capital (Latin) letters are used to denote sets while their lower case counterparts refer to the elements of sets. With some exceptions, very often the small (Latin) letters also indicate the points while their Greek (small) counterparts are used to denote the scalars, At the end, a detailed Bibliography comprising of 47 text-books (including 16 by the first author) / research monographs and 79 research papers dwelling upon the concerned topics are enlisted. However, the list is still not comprehensive. The last detail is about the first author’s own 18 other mathematical texts not directly dealing with the contents of the present presentation. The first author wishes to accord here his sincere thanks to all our teachers, especially from whom we learnt geometry and to our alma mater (Lucknow University) providing an opportunity to have developed his skills. He also thanks all the institutions both in India and abroad especially the University of Guyana, Georgetown (Guyana), Eritrea Institute of Technology, Asmara (Eritrea), Adama Science & Technology University, Adama (Ethiopia), Divine Word University, Madang (P.N.G.) allowing him to expose his talent to the students of Algebra and Topology class. Our thanks are also due to Mr. Onkar Nath Pathak, an ex- student of the first author, who took pains in reading the manuscript and made valuable suggestions. Thanks are also due to our publisher (Mr. Rohit Misra) for bringing the text to limelight. Having studied mathematics upto his first year of Masters in Mathematics he had been clever enough to understand the deeper intricacies of the subject; but (perhaps, unfortunately), having opted the more commercial subject of Management, he missed to have learned the advanced topics of Topology and Manifolds. Lucknow (India) / Semnan (Iran): June 27, 2023. Authors ______________
The Convenient Setting of Global Analysis
Mathematical Surveys and Monographs, 1997
The aim of this book is to lay foundations of differential calculus in infinite dimensions and to discuss those applications in infinite dimensional differential geometry and global analysis which do not involve Sobolev completions and fixed point theory. The approach is very simple: A mapping is called smooth if it maps smooth curves to smooth curves. All other properties are proved results and not assumptions: Like chain rule, existence and linearity of derivatives, powerful smooth uniformly boundedness theorems are available. Up to Fréchet 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.
An introduction to the completeness of compact semi-riemannian manifolds
Séminaire de théorie spectrale et géométrie, 1995
L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Séminaire de théorie spectrale et géométrie GRENOBLE 1994-1995(37-53)
Masters Project, 2020
This project is about the extension of calculus from curves and surfaces to higher dimensions. The higher-dimensional analogues of smooth curves and surfaces are called manifolds. This project was submitted for the fulfillment of my M.Sc in Mathematics degree requirements.
Aspects of the theory of infinite dimensional manifolds
Differential Geometry and its Applications, 1991
The convenient setting for smooth mappings, holomorphic mappings, and real analytic mappings in infinite dimension is sketched. Infinite dimensional manifolds are discussed with special emphasis on smooth partitions of unity and tangent vectors as derivations. Manifolds of mappings and diffeomorphisms are treated. Finally the differential structure on the inductive limits of the groups GL(n), SO(n) and some of their homogeneus spaces is treated.
Spectral asymptotics for manifolds with cylindrical ends
Annales de l’institut Fourier, 1995
L'accès aux archives de la revue « Annales de l'institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/
Analysis and Algebra on Differentiable Manifolds, 2009
Definitions 7.1.1. A locally Euclidean space is a topological space M such that each point has a neighborhood homeomorphic to an open subset of the Euclidean space R n. If ϕ is a homeomorphism of a connected open set U ⊂ M onto an open subset of R n , then U is called a coordinate neighborhood; ϕ is called a coordinate map; the functions x i = t i • ϕ, where t i denotes the ith canonical coordinate function on R n , are called the coordinate functions; and the pair (U, ϕ) (or the set (U, x 1 ,...,x n)) is called a coordinate system or a (local) chart. An atlas A of class C ∞ on a locally Euclidean space M is a collection of coordinate systems {(U α , ϕ α) : α ∈ A} satisfying the following two properties: (1) α∈A U α = M. (2) ϕ α • ϕ −1 β is C ∞ for all α, β ∈ A. A differentiable structure (or maximal atlas) F on a locally Euclidean space M is an atlas A = {(U α , ϕ α) : α ∈ A} of class C ∞ , satisfying the above two properties (1) and (2) and moreover the condition: (3) The collection F is maximal with respect to (2); that is, if (U, ϕ) is a coordinate system such that ϕ • ϕ −1 α and ϕ α • ϕ −1 are C ∞ , then (U, ϕ) ∈ F. A topological manifold of dimension n is a Hausdorff, second countable, locally Euclidean space of dimension n. A differentiable manifold of class C ∞ of dimension n (or simply differentiable manifold of dimension n, or C ∞ manifold, or n-manifold) is a pair (M, F) consisting of a topological manifold M of dimension n, together with a differentiable structure F of class C ∞ on M. The differentiable manifold (M, F) is usually denoted by M, with the understanding that when one speaks of "the differentiable manifold" M one is considering the locally Euclidean space M with some given differentiable structure F. Let M and N be differentiable manifolds, of respective dimensions m and n. A map Φ : M → N is said to be C ∞ provided that for every coordinate system (U, ϕ) on M and (V, ψ) on N, the composite map ψ • Φ • ϕ −1 is C ∞ .
Infinitesimal computations in topology
Publications mathématiques de l'IHÉS, 1977
implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/