THE THEORY OF LIE DERIVATIVES AND ITS APPLICATIONS
CONTENTS 1 DIFFERENTIABLE MANIFOLDS 1 1 1 BASIC NOTIONS AND EXAMPLES 1 1 2 LEVEL SETS 11 1 3 DIFFERENTIABLE MAPPINGS 17 1 4 TANGENT SPACE 21 1 5 TANGENT MAPPING 28 1 6 SUBMANIFOLDS 35 1 7 SUBSETS ADMITTING A SUBMANIFOLD STRUCTURE 42 1 8 TRANSVERSALITY 47 2 VECTOR BUNDLES 53 2 1 THE TANGENT BUNDLE 53 2 2 VECTOR BUNDLES 57 2 3 SECTIONS AND FRAMES 65 2 4 VECTOR BUNDLE OPERATIONS 71 2 5 TENSOR BUNDLES AND TENSOR FIELDS 78 2 6 INDUCED BUNDLES 81 2 7 SUBBUNDLES AND QUOTIENT BUNDLES 84 3 VECTOR FIELDS 93 3 1 VECTOR FIELDS AS DERIVATIONS 94 3 2 INTEGRAL CURVES AND FLOWS 97 3 3 THE LIE DERIVATIVE 109 3 4 TIME DEPENDENT VECTOR FIELDS 111 3 5 DISTRIBUTIONS AND FOLIATIONS 115 3 6 CRITICAL INTEGRAL CURVES 126 3 7 THE POINCAR MAPPING 137 3 8 STABILITY 141 3 9 INVARIANT MANIFOLDS 154 4 DIFFERENTIAL FORMS 165 4 1 BASICS 165 4 2 INTEGRATION AND INTEGRAL INVARIANTS 176 4 3 DE RHAM COHOMOLOGY 185 4 4 RIEMANNIAN MANIFOLDS 194 4 5 HODGE DUALITY 198 4 6 MAXWELL S EQUATIONS 203 4 7 PFAF AN SYSTEMS AND DIFFERENTIAL IDEALS 207 4 8 CONSTRAINTS IN CLASSICAL MECHANICS 213 5 LIE GROUPS 219 5 1 BASIC NOTIONS AND EXAMPLES 219 5 2 THE LIE ALGEBRA OF A LIE GROUP 227 5 3 THE EXPONENTIAL MAPPING 235 5 4 ADJOINT REPRESENTATION AND KILLING FORM 243 5 5 LEFT INVARIANT DIFFERENTIAL FORMS 248 5 6 LIE SUBGROUPS 256 5 7 HOMOGENEOUS SPACES 260 6 LIE GROUP ACTIONS 269 6 1 BASICS 269 6 2 KILLING VECTOR FIELDS 276 6 3 PROPER ACTIONS 281 6 4 THE TUBULAR NEIGHBOURHOOD THEOREM 286 6 5 FREE PROPER ACTIONS 290 6 6 THE ORBIT SPACE 295 6 7 INVARIANT VECTOR FIELDS 302 6 8 ON RELATIVELY CRITICAL INTEGRAL CURVES 307 7 LINEAR SYMPLECTIC ALGEBRA 315 7 1 SYMPLECTIC VECTOR SPACES 315 7 2 SUBSPACES OF A SYMPLECTIC VECTOR SPACE 318 7 3 LINEAR SYMPLECTIC REDUCTION 322 7 4 THE SYMPLECTIC GROUP 324 7 5 COMPATIBLE COMPLEX STRUCTURES 328 7 6 THE LAGRANGE GRA MANN MANIFOLD 332 7 7 THE UNIVERSAL MASLOV CLASS 338 7 8 THE KASHIWARA INDEX 346 8 SYMPLECTIC GEOMETRY 353 8 1 BASIC NOTIONS THE DARBOUX THEOREM 354 8 2 HAMILTONIAN VECTOR FIELDS AND POISSON STRUCTURES 360 8 3 THE COTANGENT BUNDLE 370 8 4 COADJOINT ORBITS 377 8 5 COISOTROPIC SUBMANIFOLDS AND CONTACT STRUCTURES 382 8 6 GENERALIZATIONS OF THE DARBOUX THEOREM 394 8 7 SYMPLECTIC REDUCTION 399 8 8 SYMPLECTOMORPHISMS AND GENERATING FUNCTIONS 403 8 9 ELEMENTARY MORSE THEORY 415 CHAPTER 1 DIFFERENTIABLE MANIFOLDS IN THIS CHAPTER WE INTRODUCE THE BASIC NOTIONS OF THE THEORY OF MANIFOLDS WE START WITH THE VERY NOTION OF A MANIFOLD ILLUSTRATE IT BY A NUMBER OF EXAMPLES AND DISCUSS THE CLASS OF EXAMPLES PROVIDED BY LEVEL SETS IN SOME DETAIL NEXT WE CARRY OVER THE NOTIONS OF DIFFERENTIABILITY THE CONCEPT OF TANGENT SPACE AND THE NOTION OF DERIVA TIVE OF A MAPPING FROM CLASSICAL CALCULUS TO THE THEORY OF MANIFOLDS IN THIS CONTEXT WE ALSO GENERALIZE THE BASIC THEOREMS OF CLASSICAL CALCULUS TO THE CASE OF MANI FOLDS FINALLY WE DISCUSS SOME MORE ADVANCED TOPICS NEEDED LATER ON THE NOTION OF SUBMANIFOLD AND THE CONCEPT OF TRANSVERSALITY SINCE THE NOTION OF A SUBMANIFOLD IS QUITE SUBTLE WE TREAT THIS SUBJECT IN DETAIL IN OUR TERMINOLOGY A SUBMANIFOLD IS DE NED BY AN INJECTIVE IMMERSION THERE ARE TWO IMPORTANT SPECIAL CLASSES OF SUB MANIFOLDS SHOWING UP IN VARIOUS APPLICATIONS THEY ARE CALLED EMBEDDED AND INITIAL RESPECTIVELY THROUGHOUT THE TEXT THE READER WILL ND A LARGE NUMBER OF ILLUSTRATIVE EXAMPLES 1 1 BASIC NOTIONS AND EXAMPLES MANIFOLDS ARE TOPOLOGICAL SPACES WHICH LOCALLY LOOK LIKE RN THEREFORE THEY ALLOW FOR AN EXTENSION OF THE NOTIONS OF CLASSICAL CALCULUS FOR THE TOPOLOGICAL NOTIONS USED IN THIS SECTION WE REFER THE READER TO THE STANDARD LITERATURE E G 53 55 199 OR 267 IN THE SEQUEL WE WILL USE THE FOLLOWING NOTATION AN ELEMENT X RN IS AN N TUPLE WRITTEN AS X X1 XN THE EUCLIDEAN SCALAR PRODUCT ON RN IS DENOTED BY X Y N I 1 XIYI AND THE CORRESPONDING NORM IS DENOTED BY FOR THE STANDARD BASIS IN RN WE WRITE E1 EN THE DUAL BASIS E 1 E N IN RN IS DE NED BY E I EJ IJ FOR X RN ITS COEF CIENTS IN THE STANDARD BASIS ARE GIVEN BY XI E I X 1 THUS WE HAVE X N I 1 XIEI OF COURSE NUMERICALLY THE NUMBERS XI AND XI COIN CIDE IF THERE IS NO DANGER OF CONFUSION WE WILL USE THE EINSTEIN SUMMATION CONVEN TION THAT IS WE WILL ALSO WRITE X XIEI DE NITION 1 1 1 TOPOLOGICAL MANIFOLD A TOPOLOGICAL SPACE M IS CALLED A TOPO LOGICAL MANIFOLD IF IT IS HAUSDORFF SECOND COUNTABLE AND LOCALLY HOMEOMORPHIC TO RN FOR SOME XED N N THIS MEANS THAT FOR EVERY M M THERE EXISTS AN OPEN NEIGHBOURHOOD U OF M IN M AND A MAPPING U RN SUCH THAT U IS OPEN IN RN AND IS A HOMEOMORPHISM ONTO ITS IMAGE THE PAIR U IS CALLED A LOCAL CHART FOR M A FAMILY A U A OF LOCAL CHARTS SATISFYING A U M IS CALLED AN ATLAS FOR M THE NUMBER N IS CALLED THE DIMENSION OF M LET U BE A LOCAL CHART ON M IF M U WE SAY THAT U IS A CHART AT M THE FUNCTIONS I E I U R 1 I N DE NE A SYSTEM OF LOCAL COORDINATES ON U THE NUMBERS I M ARE CALLED THE LOCAL COORDINATES OF M IN THE CHART U IN PARTICULAR RN IDRN IS A GLOBAL CHART FOR RN THAT IS THE IDENTICAL MAPPING ENDOWS RN WITH THE STRUCTURE OF A TOPOLOGICAL MANIFOLD THE CORRESPONDING COORDINATES ARE I X XI REMARK 1 1 2 1 SINCE RN IS NOT HOMEOMORPHIC TO RM FOR N M THE DIMENSION OF A TOPOLOGICAL MANIFOLD IS UNIQUE 2 AS A CONSEQUENCE OF BEING SECOND COUNTABLE TOPOLOGICAL MANIFOLDS CAN HAVE AT MOST COUNTABLY MANY CONNECTED COMPONENTS MOREOVER EVERY POINT POSSESSES A COUNTABLE NEIGHBOURHOOD BASIS SO THAT ONE CAN USE SEQUENCES TO TEST TOPOLOGICAL PROPERTIES LIKE CONTINUITY OF MAPPINGS OR CLOSEDNESS OF SUBSETS 3 AS A CONSEQUENCE OF BEING LOCALLY HOMEOMORPHIC TO RN TOPOLOGICAL MANIFOLDS INHERIT ALL THE LOCAL PROPERTIES OF RN THAT IS THEY ARE LOCALLY COMPACT EVERY POINT HAS A COMPACT NEIGHBOURHOOD LOCALLY CONNECTED EVERY POINT ADMITS A NEIGHBOURHOOD BASIS OF CONNECTED OPEN SETS ETC LET U1 1 AND U2 2 BE LOCAL CHARTS ON M IF U1 U2 THE MAPPING 2 1 1 RN 1 U1 U2 2 U1 U2 RN PICTURED IN FIG 1 1 AND ITS INVERSE 1 1 2 ARE CALLED THE TRANSITION MAPPINGS OF U1 1 AND U2 2 THE TRANSITION MAPPINGS ARE HOMEOMORPHISMS SINCE 1 U1 U2 AND 2 U1 U2 ARE OPEN SUBSETS OF RN IT MAKES SENSE TO ASK WHETHER THE TRANSITION MAPPINGS ARE DIFFERENTIABLE DE NITION 1 1 3 LET M BE A TOPOLOGICAL MANIFOLD AND LET K BE A NONNEGATIVE INTEGER OR THE THEORY OF LIE DERIVATIVES AND ITS APPLICATIONS 2 FIG 1 1 TWO LOCAL CHARTS AND ONE OF THEIR TRANSITION MAPPINGS 1 TWO LOCAL CHARTS U1 1 AND U2 2 ARE COMPATIBLE OF CLASS CK IF EITHER U1 U2 IS EMPTY OR THEIR TRANSITION MAPPINGS 1 1 2 AND 2 1 1 ARE OF CLASS CK 2 AN ATLAS A FOR M IS OF CLASS CK IF ANY TWO CHARTS ARE COMPATIBLE OF CLASS CK 3 TWO ATLASES A1 AND A2 OF CLASS CK ARE EQUIVALENT IF ANY CHART OF A1 IS COMPAT IBLE OF CLASS CK WITH ANY CHART OF A2 EQUIVALENTLY IF THEIR UNION IS AN ATLAS OF CLASS CK POINT 3 DE NES AN EQUIVALENCE RELATION ON THE CLASS OF ALL CK ATLASES FOR M DE NITION 1 1 4 DIFFERENTIABLE MANIFOLD AN EQUIVALENCE CLASS OF ATLASES OF CLASS CK FOR A TOPOLOGICAL MANIFOLD M IS CALLED A DIFFERENTIABLE STRUCTURE OF CLASS CK OR JUST A CK STRUCTURE ON M A TOPOLOGICAL MANIFOLD TOGETHER WITH A DIFFERENTIABLE STRUCTURE OF CLASS CK IS CALLED A DIFFERENTIABLE MANIFOLD OF CLASS CK OR JUST A CK MANIFOLD BY DE NITION A C0 MANIFOLD IS JUST A TOPOLOGICAL MANIFOLD THE PROPERTY TO BE OF CLASS C WILL BE REFERRED TO AS SMOOTH WHILE IN CHAPS 1 AND 2 THE GENERAL CK CASE IS TREATED STARTING FROM CHAP 3 EVERYTHING WILL BE ASSUMED TO BE SMOOTH WITH OCCASIONAL EXCEPTIONS IF NECESSARY REMARK 1 1 5 1 LET L K IT IS OBVIOUS THAT IF TWO LOCAL CHARTS ARE COMPATIBLE OF CLASS CK THEY ARE ALSO COMPATIBLE OF CLASS CL CONSEQUENTLY AN ATLAS OF CLASS CK IS ALSO AN ATLAS OF CLASS CL THE READER MAY CONVINCE HIMSELF THAT A CK MANIFOLD CAN BE VIEWED AS A CL MANIFOLD IF NECESSARY 2 BY ANALOGY ONE DE NES THE NOTION OF REAL AND COMPLEX ANALYTIC STRUCTURE AND MANIFOLD FOR A REAL ANALYTIC STRUCTURE THE TRANSITION MAPPINGS ARE REQUIRED TO BE REAL ANALYTIC FOR A COMPLEX ANALYTIC STRUCTURE THE LOCAL CHARTS ARE ASSUMED TO TAKE VALUES IN CN FOR SOME N AND THE TRANSITION MAPPINGS ARE ASSUMED TO BE HOLOMORPHIC THE THEORY OF LIE DERIVATIVES AND ITS APPLICATIONS 3