An Introduction to Manifolds (original) (raw)
Introduction to Differentiable Manifolds
The Physical and Mathematical Foundations of the Theory of Relativity, 2019
This is a self contained set of lecture notes. The notes were written by Rob van der Vorst. The solution manual is written by Guit-Jan Ridderbos. We follow the book 'Introduction to Smooth Manifolds' by John M. Lee as a reference text [1]. Additional reading and exercises are take from 'An introduction to manifolds' by Loring W. Tu [2].
An Introduction to Differential Geometry: The Theory of Surfaces
From a mathematical perspective, a surface is a generalization of a plane which does not necessarily require being flat, that is, the curvature is not necessarily zero. Often, a surface is defined by equations that are satisfied by some coordinates of its points. A surface may also be defined as the image, in some space of dimensions at least three, of a continuous function of two variables (some further conditions are required to insure that the image is not a curve). In this case, one says that one has a parametric surface, which is parametrized by these two variables, called parameters. Parametric equations of surfaces are often irregular at some points. This is formalized by the concept of manifold: in the context of manifolds, typically in topology and differential geometry, a surface is a manifold of dimension two; this means that a surface is a topological space such that every point has a neighborhood which is homeomorphic to an open subset of the Euclidean plane. A parametric surface is the image of an open subset of the Euclidean plane by a continuous function, in a topological space, generally a Euclidean space of dimension at least three. The paper aims at giving an introduction to the theory of surfaces from differential geometry perspective.
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 ______________
Differential Geometry: An Introduction to the Theory of Curves
Differential geometry is a discipline of mathematics that uses the techniques of calculus and linear algebra to study problems in geometry. The theory of plane, curves and surfaces in the Euclidean space formed the basis for development of differential geometry during the 18th and the 19th century. The core idea of both differential geometry and modern geometrical dynamics lies under the concept of manifold. A manifold is an abstract mathematical space, which locally resembles the spaces described by Euclidean geometry, but which globally may have a more complicated structure. The purpose of this paper is to give an elaborate introduction to the theory of curves, and those are, in general, curved. Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and in the Euclidean space by applying the concept of differential and integral calculus. The curves are represented in parametrized form and then their geometric properties and various quantities associated with them, such as curvature and arc length expressed via derivatives and integrals using the idea of vector calculus.
DIFFERENTIAL GEOMETRY: ITS PAST AND FUTURE
Differential Geometry has a long history and has been widely explored for the past more than two centuries. Recent advances in the fields of topology and abstract algebra which, by now, have undoubtedly established their dominance over almost all disciplines in pure mathematical sciences gave tremendous impetus to differential geometry. The present day differential geometry is far from the cries of Ricii’s tensor analysis initiated in the beginning of the 20th century; and can now be well regarded as differential topology. Thus, in the realms of contemporary analysis the development of the subject is relatively new and is most suitable field of pursuit. This development, however, has not been as abrupt as might be imagined from a reading of the subject. It has its roots in the movement towards differential geometry in the large to which geometers such as E. Hopf and W. Rinow, M. Cohn-Vossen, G. de Rham, W.V.D. Hodge, and S.B. Myers gave impetus. The objective of their work was to derive relationships between the topology of a manifold and its local differential geometry. Other sources of inspiration were Elié Cartan (whose fundamental contributions to exterior differentiation could be recognized by many only after his death) and M. Morse’s calculus of variations in the large. One of the major new ideas was that of a fibre bundle which gave a global structure to a differentiable manifold more general than that included in the older theories. Methods of differential geometry were applied with outstanding success to the theories of complex manifolds and algebraic varieties and those in turn have stimulated differential geometry. The discovery of invariants of the differential structure of a manifold, which are not topological invariants, by J. Milnor established differential topology as a discipline of major importance.
The Differential Geometry of Curves
Differential geometry of curves studies the properties of curves and higher-dimensional curved spaces using tools from calculus and linear algebra. This study has two aspects: the classical differential geometry which started with the beginnings of calculus and the global differential geometry which is the study of the influence of the local properties on the behavior of the entire curve. The local properties involves the properties which depend only on the behavior of the curve in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Due to this, the curves considered in differential geometry will be defined by functions which can be differentiated a certain number of times. The other aspect is the so-called global differential geometry which study the influence of the local properties on the behavior of the entire curve or surface. This paper aims to give an advanced introduction to the theory of curves, and those that are curved in general.
Dokumen.pub geometry and topology of manifolds surfaces and beyond
Preface proposed to students, who will write small dissertations, as the first author has successfully done during the years that he has delivered the master's course. The references are divided into basic reading (i.e., texts where the content of the chapter is treated in full), bibliography for the topics for further study, and references which go beyond of the content of the chapter or that have been mentioned in the text. Most of the topics of the book can be found in other texts with more specific aims. Some of the material has been written in a review form (such as homology theory, Riemannian geometry, or differential equations on manifolds). We hope that this book serves as motivation for learning all these aspects by reading deeper treatises which cover the different theories at large. Our aim is to focus in the interconnections between all these aspects. Modern geometry (from the mid-twentieth century) has seen the most important advances produced on the interaction with algebra, physics, or analysis.
1992
This document gives an additional insight into the use of the MANIFOLD system by presenting a set of non-trivial examples of programming using the MANIFOLD language. The development of these examples has been inspired by some general algorithmic patterns arising in the eld of computer graphics and the use of computing farms. The document presupposes that the reader is familiar with the syntax and the semantics of MANIFOLD.
The notion of abstract manifold: a pedagogical approach
arXiv: Mathematical Physics, 2012
A self-contained introduction is presented of the notion of the (abstract) dierentiable manifold and its tangent vector elds. The way in which elementary topological ideas stimulated the passage from Euclidean (vector) spaces and linear maps to abstract spaces (manifolds) and dieomorphisms is emphasized. Necessary topological ideas are introduced at the beginning in order to keep the text as self-contained as possible. Connectedness is presupposed in the denition of the
DISCLAIMER: This paper is a REJECTED submission for PME (2010). I'm just uploading it due to the similarity (in fact, just a translation from Protuguese) with another publicated paper.
Analysis in Vector Spaces: A Course in Advanced Calculus
2009
Diffeomorphisms are invertible mappings between two open sets that are continuously differentiable in both directions. The simplest type of diffeomorphism is an invertible linear transformation. This is a basic concept in linear algebra. Similarly, the study of diffeomorphisms is a basic part of differential calculus. Most of the results on diffeomorphisms depend on the inverse function theorem. This is one of the central results we obtain in this course. It states that if a continuously differentiable function has an invertible derivative at a point, then its restriction to a neighborhood of that point is a diffeomorphism. We also call a continuously differentiable function a C 1 function. Diffeomorphisms and the inverse function theorem are the gateway to the study of manifolds: surfaces, curves and other lower-dimensional structures that are embedded in a larger space. A good example is the upper half of the surface of the unit sphere in M 3 , which is a two-dimensional manifold. We can represent this set as the image of the unit disk { (x, y) | x 2 + y 2 < 1 } in M 2 under the mapping Analysis in Vector Spaces.