Shahshahani - An Introductory Course on Differentiable Manifolds

Copyright

Copyright 2016 by Siavash Shahshahani

All rights reserved.

Bibliographical Note

An Introductory Course on Differentiable Manifolds is a new work, first published by Dover Publications, Inc., in 2016, as part of the Aurora Dover Modern Math Originals series.

International Standard Book Number

ISBN-13: 978-0-486-80706-5

ISBN-10: 0-486-80706-1

Manufactured in the United States by RR Donnelley

80706101 2016

www.doverpublications.com

in memory of my parents

This book is the outgrowth of a course on differentiable manifolds that the author has taught many times over the years. The audience has mostly consisted of advanced undergraduate and first-year graduate students in mathematics. A course on manifolds has now become a staple of solid education in mathematics not only for mathematicians, but also for many theoretical physicists and others. One could argue that such a course is the natural modern day setting for doing advanced calculus on a curved space. The material in this book is slightly more than the author has been able to cover in a one-semester course for the intended audience, but a two-quarter course should allow a leisurely paced coverage. Alternatively, various choices of omission are available for a one-semester course after covering most of the first seven chapters.

The prerequisite for this course is solid grounding in undergraduate mathematics, including rigorous multivariable calculus, linear algebra, elementary abstract algebra and point set topology. In practice, we have found that the ideal background is often lacking, so we have not shied away from recalling some of the material either in the text or in the appendices. We have also strived to make the treatment complete, providing sufficient detail for the novice so that they will be able to gain confident mastery of the material. In particular, we have made a point of not leaving any details other than the completely routine or repetitive to the reader, nor have we relegated any crucial argument in the text material to the exercises. In the same vein, we have avoided the use of time-worn phrases such as it is obvious..., believing that the obvious should best be passed by in silence. On the other hand, over 250 exercises, tuned to the text material, should offer students sufficient opportunity to gauge their skills and to gain additional depth and insight.

An explanation of the arrangement of material in this book is in order. In the earliest lecture note versions of the book, we followed the familiar sequence of manifolds-vector bundles-vector fields and tensor fields, as a starter. In the present work, based on our recent teaching practice, we have divided the book into four parts. , Geometric Structures, consisting of the single final chapter of the book, we provide a glimpse into geometric structures through the introduction of connections on the tangent bundle as a tool to implant the second derivative and the derivative of vector fields on the base manifold. In the early versions of the work, we concluded with a chapter on symplectic geometry and Hamiltonian mechanics. But symplectic geometry is now witnessing such rapid and broad development that we thought it wise to forego such an introduction and replace Hamiltonian mechanics with a purely Newtonian account tied closely to the notions of connection and acceleration vector as developed in the final chapter.

One may cite logical, pedagogical and even historical justification for this choice of presentation. While the appearance of the word manifold (Mannigfaltigkeit) in mathematics literature predates the introduction of topological spaces by at least half a century, the formal definition of a manifold had to await the development of point set theory. Before that happened, many tools of differential geometry were developed locally in tensor language. The pioneers used coordinates and local computations while differentiating between pure analysis and geometry by requiring that geomeric entities be held to certain rigid transformations under coordinate change. The student who is initially presented with the whole apparatus of manifolds and vector bundles will often need plenty of time to sort out the logical niche of various notions and frequently experiences difficulty working out examples and making computations. Our experience has been that by first developing the algebraic and local computational skills of students, and by treating what is genuinely local as local, we can better prepare the student to appreciate the necessity and usefulness of global notions and the facility they provide in dealing directly and intrinsically with geometric notions. Even the integration of plane fields (Frobenius theorem) and the notion of connection could have been included in the early chapters on local theory, but the significant global aspects of these notions make it more expedient to present them after global tools have been developed.

The editorial organization of the book is as follows. Each chapter is divided into sections serialized alphabetically by capital letters A, B, C,... Each section is divided into subsections identified by numbers. These numbers run consecutively not by section, but by the chapter they are in. A subsection may be a significant theorem, an important definition, a propostion describing elementary consequences of a definition, a set of examples or even a heuristic discussion. Further, some subsections are subdivided into sub-subsections serially identified by small letters a, b, c,... Not many statements are honored as theorems throughout the book. Most assertions requiring proofs are identified by a subsection number or a small letter indicating a sub-subsection. The dearth of serious theorems is natural as this is only an introductory text aimed at providing necessary language skills.

I have had the fortune of being associated with highly talented students during repeated presentations of the course on which this book is based. Their spirited interaction has sharpened and reshaped the original notes. Many of these former students are now accomplished mathematicians. I will refrain from naming many, lest I inadvertently leave out some equally deserving. But I will mention a few, in chronological order, who gave me written suggestions I happen to have kept: F. Rezakhanloo, A.S. Tahvildarzadeh, H. Torabi Tehrani, B. Khanedani, A. Taghavi and S. Zakeri. The latter has also been, more recently, my ever generous TeX advisor. The finished manuscript was improved by comments and suggestions from A. Taghavi, S. Habibi-Esfahani, M. Shahriari, M. Khoshnevis and M. R. Koushesh. Other mathematicians whose contributions I should recognize include A. Shafii-Dehabad, M. Ardeshir, A. Jafari, P. Safari, M. Zeinalian and Sohrab Shahshahani. Most of all, Bahman Khanedani has been a constant mathematical companion and teaching colleague from whom I have learned much; his footprint is all over the book.

Special thanks are due to A. Kamalinejad who initiated the TeX version of the book. I have benefited in this and earlier work from the expert graphical skill of Amin Sadeghi whose tutelage also launched M. Mazaheri into generously giving many hours of his time to preparing and importing the diagrams. The final English version of the book was prepared in New York where I had access to the facilities of CUNY Graduate Center. I thank Dennis Sullivan for the opportunity.

S. Shahshahani

December 2015

Siavash Shahshahani received his university education at Berkeley in the 1960s, getting a PhD in mathematics under Steve Smale in 1969. Subsequently he held positions at Northwestern and the University of Wisconsin, Madison. From 1974 to his retirement in 2012, he was mainly at Sharif University of Technology, Tehran, Iran, and helped develop a strong mathematics program there.