Differential manifolds : a basic approach for experimental physicists

593 Pages·2014·21.777 MB·English

by  Baillon, Paul

Differential Manifolds
A Basic Approach for Experimental Physicists
World Scientific
Differential Manifolds
A Basic Approach for Experimental Physicists
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Differential Manifolds
A Basic Approach for Experimental Physicists
Paul Baillon
CERN, Switzerland
¥ World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Library of Congress Cataloging-in-Publication Data
Baillon, Paul.
Differential manifolds : a basic approach for experimental physicists / by Paul Baillon (CERN,
Switzerland),
pages cm
Includes bibliographical references and index.
ISBN 978-981-4449-56-4 (hardcover : alk. paper)
1. Differentiable manifolds. 2. Manifolds (Mathematics) I. Title.
QA614.3.B35 2013
516.3'6—dc23
2013010435
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(CC BY-SA 2.5)
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Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
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Contents
Preface ix
1 Manifold 1
1.1 Differentiable manifold...................................................................................................... 1
1.2 Smooth maps.................................................................................................................... 3
1.3 Vector fields on a differentiable manifold..................................................................... 8
1.3.1 Definition of a vector field................................................................................ 8
1.3.2 Properties of vector fields ................................................................................ 9
1.4 Conventions........................................................................................................................ 19
1.5 Tangent spaces and tangent vectors............................................................................ 19
1.6 Coordinate changes......................................................................................................... 23
1.7 Metric on a differentiable manifold................................................................................ 25
1.8 One-form field and differential....................................................................................... 32
1.9 Tensorial field.................................................................................................................... 40
1.10 Wedge product of 1-linear forms (versus vector fields)............................................ 45
1.11 Exterior differential......................................................................................................... 49
1.12 Volume and integral in differential manifold.............................................................. 54
1.12.1 Volume form......................................................................................................... 54
1.12.2 Oriented integral.................................................................................................. 61
1.12.3 Convex differentiable manifold......................................................................... 67
1.12.4 Stokes Kelvin-Thomson theorem on a convex smooth surface.................. 74
1.13 Lie bracket ........................................................................................................................ 85
1.14 Bundles and differentiable manifold............................................................................ 86
1.14.1 Bundles and sections of vector bundle.......................................................... 86
1.14.2 Covariant derivative........................................................................................... 90
1.14.3 Vector potential.................................................................................................. 91
1.14.4 Differential on a bundle .................................................................................... 96
1.14.5 Exterior differential on a bundle..................................................................... 96
1.15 Parallel transport............................................................................................................. 98
1.16 Curvature............................................................................................................................ 98
1.17 Lagrangian of the electro-weak interactions.................................................................101
1.17.1 Applications............................................................................................................105
1.18 General relativity................................................................................................................108
1.18.1 Einstein equations in vacuum ............................................................................118
1.19 Notations...............................................................................................................................128
2 Some basic mathematics needed for manifolds 131
2.1 General concepts ...............................................................................................................131
2.1.1 Maps and functions ..............................................................................................131
v
VI CONTENTS
2.1.2 Topology....................................................................................................................151
2.1.3 Continuous m ap.....................................................................................................165
2.2 Real numbers, set 72.............................................................................................................170
2.2.1 Example of a real number...................................................................................171
2.2.2 Order on 72.............................................................................................................171
2.2.3 Operations on 72......................................................................................................173
2.2.4 Upper Jimit and lower Jimit of a subset of 72................................................174
2.3 Euclidean Metric ................................................................................................................176
2.4 Metric and topology on 72..................................................................................................181
2.5 Behavior at a point............................................................................................................182
2.6 Some properties of continuous maps from 72. to 72........................................................185
2.6.1 Distributivity of product with the sum.............................................................186
2.6.2 Commutativity of sum and product ................................................................187
2.6.3 Associativity of sum and product........................................................................188
2.7 Continuous maps from topological sets to 72...................................................................188
2.8 Derivable function................................................................................................................192
2.9 Group......................................................................................................................................205
2.9.1 Finite groups............................................................................................................207
2.9.2 Operations on groups over a finite set of index................................................211
2.9.3 Rings, fields............................................................................................................214
2.10 Module over a commutative ring......................................................................................218
2.10.1 Wedge projection.....................................................................................................232
2.11 Vector spaces.......................................................................................................................235
2.11.1 Definition and properties of vector space......................................................235
2.11.2 Linear forms............................................................................................................244
2.11.3 Multilinear forms.....................................................................................................246
2.11.4 Wedge projection, antisymmetric multilinear forms....................................247
2.11.5 Tensorial product and linear transformation....................................................250
2.11.6 Riemannian inner product (or simply inner product).................................259
2.12 72." 266
2.13 Complex numbers................................................................................................................271
2.13.1 Properties of complex numbers...........................................................................272
2.13.2 Semi Riemannian metric......................................................................................273
2.14 Convex subset.......................................................................................................................277
2.15 Topology on 72."...................................................................................................................285
2.15.1 Projection of 72." on 72."-1 ...................................................................................289
2.15.2 Compact set in 72.” .................................................................................................290
2.16 Continuous map on 72.n to 72.p..........................................................................................296
2.17 Sequence..................................................................................................................................297
2.18 Sequence in 72.,^,...................................................................................................................301
2.19 Sequence of maps................................................................................................................302
2.20 Partial derivative................................................................................................................304
2.20.1 Derivative of a sequence of maps.........................................................................311
2.21 Topology on convex subsets..............................................................................................312
2.22 Path connected sets............................................................................................................326
2.23 Riemann integral of maps with compact support.........................................................331
2.24 Volume in 72.".......................................................................................................................341
2.25 Integral of a continuous map..............................................................................................352
2.25.1 Some properties of integrals on an interval of 72.........................................366
CONTENTS vii
2.26 Differential equations........................................................................................................369
2.27 Lebesgue integral................................................................................................................377
2.28 Taylor expansion of functions with derivatives.............................................................392
2.28.1 Remarks on complex functions...........................................................................403
2.29 Exponentials...........................................................................................................................405
2.29.1 Exponential of complex numbers........................................................................411
2.29.2 Trigonometry............................................................................................................413
2.30 Polynomials...........................................................................................................................416
2.31 Useful smooth maps built with exponentials................................................................424
2.32 Eigenvectors of a linear transformation ........................................................................428
2.32.1 Eigenvectors for a self adjoint linear transformation...............................429
A Conventions, basic relations and symbols 433
A.l Logic theory..........................................................................................................................433
A. 1.1 Logic symbols........................................................................................................433
A. 1.2 Demonstration........................................................................................................434
A. 1.3 Properties of logical symbols...............................................................................435
A. 1.4 Deduced properties of logical symbols..........................................................435
A. 1.5 Substitution criteria.............................................................................................440
A.2 Specifics terms......................................................................................................................440
A.2.1 Terms associated with a relation........................................................................440
A.3 Quantificators......................................................................................................................441
A.4 Specifics relations...............................................................................................................445
A.4.1 Functional relations .............................................................................................445
A.4.2 Equivalence relation.............................................................................................445
A.4.3 Representative of an equivalence relation.........................................................446
A.4.4 Ordering relation.....................................................................................................446
A.5 Sets........................................................................................................................................447
A.5.1 Symbols of the set theory ...................................................................................447
A.5.2 Subsets.......................................................................................................................451
A.5.3 Union and intersection of sets...........................................................................453
A.5.4 Complement of a subset......................................................................................454
A.5.5 Product of two sets.................................................................................................458
A.5.6 Union and intersection of sets over a set of indices....................................462
A.5.7 Equivalence relation on sets...............................................................................468
A.5.8 Product of sets over a set of indices ..............................................................473
A.5.9 Order on sets............................................................................................................482
A.5.10 Cardinal...................................................................................................................499
A.6 Integers.................................................................................................................................516
A.6.1 Recurrence principle..............................................................................................520
A.6.2 Calculus on integer.................................................................................................521
A.7 Operations on Z = Z0+ U Z ~ .....................................................................................532
A.8 Rational numbers...............................................................................................................551
A.8.1 Notation commonly used with rational numbers...........................................552
A.8.2 Operations on rational numbers........................................................................553
A.8.3 Order on Q ............................................................................................................557
A.9 Conventions..........................................................................................................................560
Bibliography 561
viii CONTENTS
List of definitions 563
List of conventions 573
Index 575
Preface
Differential manifold is now a major frame for describing our physical world. There axe several
lecture notes on the subject dedicated to physicists. Those books, though they give a complete
view of the subject, they often neglect to give the precise mathematical frame in which they
do their development.
They just accept important theorems and think probably that physicists do not have enough
background in mathematics for them. Very often for less important propositions they let to
the reader the burden of finding a correct proof. For non-experts, it is a problem because
those propositions are sometimes not that easy to prove. Nevertheless in spite of those critics,
I am grateful to those lecture notes because they are the sources of my book on differential
manifold [2], [3]. They also provide an important bibliography on the subject.
On the other hand, lecture notes for mathematicians on the same subject are unfortunately
very obscure for non experts.
This book was written to fill the gap between those two kinds of lecture notes. It is aimed
for students in physics and to physicists who want to enter in the framework of differential
Manifold.
Differential manifolds deal with continuous maps which are indefinitely derivable. They deal
also with integrals, differential equations and change of variables acting on various topological
spaces. For that you need a very precise description of the various mathematical items used.
That is why this book includes a description of standard general mathematics limited to what
is needed for the differential manifolds. In order to have a precise language, you need a precise
logical framework. We include it also in the book. It comes mainly from [1]. Strange enough
when limited to what is really needed, those mathematics are not difficult and it is much easier
to understand in a solid mathematical frame than in an artistic fuzzy.
I want to thank Robert Barate, Raymond Stora and Daniel Treille, for the time that they
spent to read this book and for their advices and corrections. I also thank Philippe Bloch for
his encouragements and comments. I am very grateful to CERN for its hospitality.
IX