this book grew out of courses given at the instituto de fisica teorica for many years. as the title announces, it is intended as a first, elementary approach to "geometrical physics" -- to be understood as a chapter of mathematical physics. mathematical physics is a moving subject, and has moved faster in recent times. from the study of differential equations and related special functions, it has migrated to the more qualitative realms of topology and algebra. the bridge has been the framework of geometry. the passage supposes an acquaintance with concepts and terms of a new kind, to which this text is a tentative introduction. in its technical uses, the word "geometry" has since long lost its metric etymological meaning. it is the science of space, or better, of spaces. thus, the name should be understood as a study of those spaces which are of interest in physics. this emphasis on the notion of space has dominated the choice of topics - they will have in common the use of "spaces". some may seem less geometric than others, but a space is always endowed with a few basic, irreducible properties enabling some kind of analysis, allowing a discussion of relations between its different parts.
目录
0SPACE AND GEOMETRY PARTⅠMANIFOLDS 1 GENERAL TOPOLOGY 1.0 INTRODUCTORY COMMENTS 1.1 TOPOLOGICAL SPACES 1.2 KINDS OF TEXTURE 1.3 FUNCTIONS 1.4 QUOTIENTS AND GROUPS 1.4.a Quotient spaces 1.4.b Topological groups 2 HOMOLOGY 2.1 GRAPHS 2.1.a Graphs, first way 2.1.b Graphs, second way 2.2 THE FIRST TOPOLOGICAL INVARIANTS 2.2.a Simplexes, complexes & all that 2.2.b Topological numbers 3 HOMOTOPY 3.0 GENERAL HOMOTOPY 3.1 PATH HOMOTOPY 3.1.a Homotopy of curves 3.1.b The Fundamental group 3.1.c Some calculations 3.2 COVERING SPACES 3.2.a Multiply-connected Spaces 3.2.b Coveting Spaces 3.3 HIGHER HOMOTOPY 4 MANIFOLDS & CHARTS 4.1 MANIFOLDS 4.1.a Topological manifolds 4.1.b Dimensions, integer and other 4.2 CHARTS AND COORDINATES 5 DIFFERENTIABLE MANIFOLDS 5.1 DEFINITION AND OVERLOOK 5.2 SMOOTH FUNCTIONS 5.3 DIFFERENTIABLE SUBMANIFOLDS PARTⅡDIFFERENTIABLE STRUCTURE 6 TANGENT STRUCTURE 6.1 INTRODUCTION 6.2 TANGENT SPACES 6.3 TENSORS ON MANIFOLDS 6.4 FIELDS & TRANSFORMATIONS 6.4.a Fields 6.4.b Transformations 6.5 FRAMES 6.6 METRIC & RIEMANNIAN MANIFOLDS 7 DIFFERENTIAL FORMS 7.1 INTRODUCTION 7.2 EXTERIOR DERIVATIVE 7.3 VECTOR-VALUED FORMS 7.4 DUALITY AND CODERIVATION 7.5 INTEGRATION AND HOMOLOGY 7.5.a Integration 7.5.b Cohomology of differential forms 7.6 ALGEBRAS, ENDOMORPHISMS AND DERIVATIVES 8 SYMMETRIES 8.1 LIE GROUPS 8.2 TRANSFORMATIONS ON MANIFOLDS 8.3 LIE ALGEBRA OF A LIE GROUP 8.4 THE ADJOINT REPRESENTATION 9 FIBER BUNDLES 9.1 INTRODUCTION 9.2 VECTOR BUNDLES 9.3 THE BUNDLE OF LINEAR FRAMES 9.4 LINEAR CONNECTIONS 9.5 PRINCIPAL BUNDLES 9.6 GENERAL CONNECTIONS 9.7 BUNDLE CLASSIFICATION PARTⅢFINAL TOUCH 10 NONCOMMUTATIVE GEOMETRY 10.1 QUANTUM GROUPS -- A PEDESTRIAN OUTLINE 10.2 QUANTUM GEOMETRY PARTⅣ MATHEMATICAL TOPICS PARTⅤPHYSICAL TOPICS GLOSSARY REFERENCES ALPHABETIC INDEX
