HANDBOOK OF GLOBAL ANALYSIS
CONTENTS GLOBAL ASPECTS OF FINSLER GEOMETRY FINSLER METRICS AND CONNECTIONS GEODESICS IN FINSLER MANIFOLDS COMPARISON THEOREMS: CARTAN–HADAMARD THEOREM, BONNET–MYERS THEO- REM, LAPLACIAN AND VOLUME COMPARISON RIGIDITY THEOREMS: FINSLER MANIFOLDS OF SCALAR CURVATURE AND LOCALLY SYM- METRIC FINSLER METRICS CLOSED GEODESICS ON FINSLER MANIFOLDS, SPHERE THEOREM AND THE GAUSS– BONNET FORMULA MORSE THEORY AND NONLINEAR DIFFERENTIAL EQUATIONS INTRODUCTION GLOBAL THEORY LOCAL THEORY APPLICATIONS STRONGLY INDE?NITE MORSE THEORY STRONGLY INDE?NITE VARIATIONAL PROBLEMS INDEX THEORY INTRODUCTION AND SOME HISTORY FREDHOLM OPERATORS – THEORY AND EXAMPLES THE SPACE OF FREDHOLM OPERATORS AND K-THEORY ELLIPTIC OPERATORS AND SOBOLEV SPACES THE ATIYAH-SINGER INDEX THEOREM GENERALIZATIONS PARTIAL DIFFERENTIAL EQUATIONS ON CLOSED AND OPEN MANIFOLDS SOBOLEV SPACES NON-LINEAR SOBOLEV STRUCTURES SELF-ADJOINT LINEAR DIFFERENTIAL OPERATORS ON MANIFOLDS AND THEIR SPECTRAL THEORY THE SPECTRAL VALUE ZERO THE HEAT EQUATION, THE HEAT KERNEL AND THE HEAT ?OW THE WAVE EQUATION, ITS HAMILTONIAN APPROACH AND COMPLETENESS INDEX THEORY ON OPEN MANIFOLDS THE CONTINUITY METHOD FOR NON-LINEAR PDES ON OPEN MANIFOLDS TEICHM¨ULLER THEORY HARMONIC MAPS NON-LINEAR ?ELD THEORIES GAUGE THEORY FLUID DYNAMICS THE RICCI ?OW SPECTRAL GEOMETRY THE GEOMETRY OF OPERATORS OF LAPLACE AND DIRAC TYPE HEAT TRACE ASYMPTOTICS FOR CLOSED MANIFOLDS HEARING THE SHAPE OF A DRUM HEAT TRACE ASYMPTOTICS OF MANIFOLDS WITH BOUNDARY HEAT TRACE ASYMPTOTICS AND INDEX THEORY HEAT CONTENT ASYMPTOTICS HEAT CONTENT WITH SOURCE TERMS TIME DEPENDENT PHENOMENA SPECTRAL BOUNDARY CONDITIONS OPERATORS WHICH ARE NOT OF LAPLACE TYPE THE SPECTRAL GEOMETRY OF RIEMANNIAN SUBMERSIONS LAGRANGIAN FORMALISM ON GRASSMANN MANIFOLDS GRASSMANN MANIFOLDS THE LAGRANGIAN FORMALISM ON A GRASSMANN MANIFOLD LAGRANGIAN FORMALISM ON SECOND ORDER GRASSMANN BUNDLES SOME APPLICATIONS SOBOLEV SPACES ON MANIFOLDS MOTIVATIONS SOBOLEV SPACES ON MANIFOLDS: DE?NITION AND ?RST PROPERTIES EQUALITY AND DENSITY ISSUES EMBEDDING THEOREMS, PART I EUCLIDEAN TYPE INEQUALITIES EMBEDDING THEOREMS, PART II EMBEDDING THEOREMS, PART III COMPACT EMBEDDINGS BEST CONSTANTS EXPLICIT SHARP INEQUALITIES THE CARTAN-HADAMARD CONJECTURE HARMONIC FUNCTIONS ON EUCLIDEAN SPACES HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS WEAKLY HARMONIC MAPS AND SOBOLEV SPACES BETWEEN MANIFOLDS REGULARITY EXISTENCE METHODS OTHER ANALYTICAL PROPERTIES TWISTOR THEORY AND COMPLETELY INTEGRABLE SYSTEMS TOPOLOGY OF DIFFERENTIABLE MAPPINGS MANIFOLDS AND SINGULARITIES MILNOR ?BRE MONODROMY STRATI?CATIONS OF SPACES STRATI?ED MORSE THEORY RECTI?ED HOMOTOPICAL DEPTH RELATIVE STRATI?ED MORSE THEORY TOPOLOGY OF IMAGES AND MULTIPLE POINT SPACES THE IMAGE COMPUTING SPECTRAL SEQUENCE GROUP ACTIONS AND HILBERT’S ?FTH PROBLEM HILBERT’S ?FTH PROBLEM LIE GROUPS AND MANIFOLDS GROUP ACTIONS CARTAN AND PROPER ACTIONS OF LIE GROUPS NON-PARACOMPACT CARTAN G-MANIFOLDS HOMOGENEOUS SPACES OF LIE GROUPS TWISTED PRODUCTS SLICES THE STRONG CR TOPOLOGIES, 1 ?R < ?, AND THE VERY-STRONG C?TOPOLOGY CONTINUITY OF INDUCED MAPS IN THE STRONG CR TOPOLOGIES, 1 ?R < ?, AND IN THE VERY-STRONG C?TOPOLOGY THE PRODUCT THEOREM THE EQUIVARIANT GLUEING LEMMA WHITNEY APPROXIMATION HAAR INTEGRALS OF CS MAPS, 1 ?S ??, AND OF REAL ANALYTIC MAPS CONTINUITY OF THE AVERAGING MAPS IN THE STRONG CR TOPOLOGIES, 1 ?R < ?, AND IN THE VERY-STRONG C?TOPOLOGY APPROXIMATION OF K-EQUIVARIANT CS MAPS, 1 ?S ??, BY K- EQUIVARIANT REAL ANALYTIC MAPS, IN THE STRONG CS TOPOLOGIES, 1 ?S < ?, AND IN THE VERY-STRONG C?TOPOLOGY APPROXIMATION OF CS K-SLICES, 1 ?S ?? PROOF OF THE MAIN THEOREM EXTERIOR DIFFERENTIAL SYSTEMS NIKY KAMRAN BASIC EXISTENCE THEOREMS FOR INTEGRAL MANIFOLDS OF C?SYSTEMS INVOLUTIVE ANALYTIC SYSTEMS AND THE CARTAN-K¨AHLER THEOREM PROLONGATION AND THE CARTAN-KURANISHI THEOREM A CARTAN-K¨AHLER THEOREM FOR C?PFAF?AN SYSTEMS CHARACTERISTIC COHOMOLOGY TOPOLOGICAL OBSTRUCTIONS APPLICATIONS TO SECOND-ORDER SCALAR HYPERBOLIC PARTIAL DIFFERENTIAL EQUA- TIONS IN THE PLANE SOME APPLICATIONS TO DIFFERENTIAL GEOMETRY WEIL BUNDLES AS GENERALIZED JET SPACES WEIL ALGEBRAS WEIL BUNDLES ON THE GEOMETRY OF T A-PROLONGATIONS FIBER PRODUCT PRESERVING BUNDLE FUNCTORS DISTRIBUTIONS, VECTOR DISTRIBUTIONS, AND IMMERSIONS OF MANIFOLDS IN EUCLIDEAN SPACES DISTRIBUTIONS ON EUCLIDEAN SPACES DISTRIBUTIONS AND RELATED CONCEPTS ON MANIFOLDS VECTOR DISTRIBUTIONS OR PLANE ?ELDS ON MANIFOLDS IMMERSIONS AND EMBEDDINGS OF MANIFOLDS IN EUCLIDEAN SPACES GEOMETRY OF DIFFERENTIAL EQUATIONS GEOMETRY OF JET SPACES ALGEBRA OF DIFFERENTIAL OPERATORS FORMAL THEORY OF PDES LOCAL AND GLOBAL ASPECTS GLOBAL VARIATIONAL THEORY IN ?BRED SPACES PROLONGATIONS OF ?BRED MANIFOLDS DIFFERENTIAL FORMS ON PROLONGATIONS OF ?BRED MANIFOLDS LAGRANGE STRUCTURES THE STRUCTURE OF THE EULER-LAGRANGE MAPPING INVARIANT VARIATIONAL PRINCIPLES REMARKS SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS IN JET BUNDLES AND THE INVERSE PROBLEM OF THE CALCULUS OF VARIATIONS SECOND-ORDER DIFFERENTIAL EQUATIONS ON ?BRED MANIFOLDS VARIATIONAL STRUCTURES IN THE THEORY OF SODES SYMMETRIES AND ?RST INTEGRALS GEOMETRY OF REGULAR SODES ON R × TM THE INVERSE PROBLEM FOR SEMISPRAYS ELEMENTS OF NONCOMMUTATIVE GEOMETRY ALGEBRAS INSTEAD OF SPACES MODULES AS BUNDLES HOMOLOGY AND COHOMOLOGY THE CHERN CHARACTERS CONNECTIONS AND GAUGE TRANSFORMATIONS NONCOMMUTATIVE MANIFOLDS TORIC NONCOMMUTATIVE MANIFOLDS THE SPECTRAL GEOMETRY OF THE QUANTUM GROUP SUQ(2) DE RHAM COHOMOLOGY DE RHAM COMPLEX INTEGRATION AND DE RHAM COHOMOLOGY DE RHAM CURRENTS HARMONIC FORMS GENERALIZATIONS OF THE DE RHAM COMPLEX EQUIVARIANT DE RHAM COHOMOLOGY COMPLEXES OF DIFFERENTIAL FORMS ASSOCIATED TO DIFFERENTIAL GEOMETRIC STRUCTURES TOPOLOGY OF MANIFOLDS WITH CORNERS QUADRANTS DIFFERENTIATION THEORIES MANIFOLDS WITH CORNERS MANIFOLDS WITH GENERALIZED BOUNDARY JET MANIFOLDS AND NATURAL BUNDLES JETS DIFFERENTIAL EQUATIONS THE CALCULUS OF VARIATIONS NATURAL BUNDLES SOME ASPECTS OF DIFFERENTIAL THEORIES BACKGROUND CALCULUS IN TOPOLOGICAL VECTOR SPACES AND BEYOND THE CHERN – RUND DERIVATIVE VARIATIONAL SEQUENCES PRELIMINARIES CONTACT FORMS VARIATIONAL BICOMPLEX AND VARIATIONAL SEQUENCE C-SPECTRAL SEQUENCE AND VARIATIONAL SEQUENCE FINITE ORDER VARIATIONAL SEQUENCE SPECIAL TOPICS NOTES ON THE DEVELOPMENT OF THE SUBJECT THE OKA-GRAUERT-GROMOV PRINCIPLE FOR HOLOMORPHIC BUNDLES STEIN MANIFOLDS AND STEIN SPACES OKA’S THEOREM GRAUERT’S OKA PRINCIPLE GROMOV’S OKA PRINCIPLE THE CASE OF RIEMANN SURFACES COMPLETE INTERSECTIONS EMBEDDING DIMENSIONS OF STEIN SPACES OKA PRINCIPLE WITH GROWTH CONDITION OKA’S PRINCIPLE AND THE MOVING LEMMA IN HYPERBOLIC GEOMETRY THE ALGEBRAIC VERSION OF OKA’S PRINCIPLE COMPARISON THEOREMS: CARTAN-HADAMARD THEOREM, BONNET-MYERS THEOREM, LAPLACIAN AND VOLUME COMPARISON RIGIDITY THEOREMS: FINSLER MANIFOLDS OF SCALAR CURVATURE AND LOCALLY SYMMETRIC FINSLER METRICS CLOSED GEODESICS ON FINSLER MANIFOLDS, SPHERE THEOREM AND THE GAUSS-BONNET FORMULA FINSLER METRICS LET ? : TM ?M BE THE TANGENT BUNDLE OF A CONNECTED SMOOTH MANIFOLD M OF DIM M = N WE DENOTE BY V = (X, Y) THE POINTS IN TM IF Y ???1(X) = TXM WE DENOTE BY Z(M) THE ZERO SECTION OF TM, AND BY TM × THE SLIT TANGENT BUNDLE TM\Z(M) WE INTRODUCE A COORDINATE SYSTEM ON TM AS FOLLOWS LET U ?M BE AN OPEN SET WITH LOCAL COORDINATE (X1, · · · , XN) BY SETTING V = P YI ?/?XI X FOR EVERY V ???1(U), WE INTRODUCE A LOCAL COORDINATE (X, Y) = (X1, · · · , XN, Y1, · · · , YN) ON ??1(U) DE?NITION 11 A FUNCTION F : TM ??R IS CALLED A FINSLER METRIC ON M IF (1) F(X, Y) ?0, AND F(X, Y) = 0 IF AND ONLY IF Y = 0, (2) F(X, ?Y) = ?F(X, Y) FOR ?? ?R+ = {? ?R : ? > 0}, (3) F(X, Y) IS SMOOTH ON TM ×, THE OUT-SIDE OF THE ZERO SECTION, (4) G = F 2/2 IS STRICTLY CONVEX ON EACH TANGENT SPACE TXM, THAT IS, THE HESSIAN (GIJ) DE?NED BY GIJ(X, Y) = ?2G ?YI?YJ (11) IS POSITIVE-DE?NITE, ARE SATIS?ED THE PAIR (M, F) IS CALLED A FINSLER MANIFOLD WE NOTE THAT THE LAST CONDITION IN THIS DE?NITION IS EQUIVALENT TO THE CONVEXITY OF THE UNIT BALL BX = {Y ?TXM | F(X, Y) ?1} IF A FINSLER METRIC F IS DE?NED, THEN THE NORM ?Y?OF EACH Y ?TXM IS DE?NED BY ?Y?= F(X, Y), AND THE LENGTH S(T) OF A SMOOTH CURVE C(T) = (X1(T), · · · , XN(T)) IS DE?NED BY S(T) = Z ??C(T)?DT = Z 2D LOG ? THE NORM ???H OF ? WITH RESPECT TO GH IS GIVEN BY ??(X)?H = ?X?< 1, AND THUS THE FUNCTION F ON B DE?NED BY F(V) = P GH(V, V) + ?(V) IS A FINSLER METRIC CALLED THE FUNK METRIC ON B WE NOTE THAT THE RELATION BETWEEN GH AND F IS GIVEN BY ?V?H = 2 [F(V) + F(?V)] FOR ALL V ?TM FOR THE DIFFERENTIAL ??OF THE SUBMERSION ? : TM × ??M, THE VERTICAL SUBBUNDLE V OF T(TM ×) IS DE?NED BY V = KER ??, AND V IS LOCALLY SPANNED BY {?/?Y1, · · · , ?/?YN} ON EACH ??1(U) THEN IT INDUCES THE EXACT SEQUENCE 0 ??V I ??T(TM ×) ?? ??G TM ??0, (12) WHERE G TM = {(Y, V) ?TM × × TM V ?T?(Y)M} IS THE PULL-BACK BUNDLE ??TM TM × G TM TM M - ? - ? ? ? HANDBOOK OF GLOBAL ANALYSIS SINCE THE NATURAL LOCAL FRAME ?ELD {?/?XI}I=1,··· ,N ON EACH U IS IDENTI?ED WITH THE ONE OF G TM ON ??1(U), ANY SECTION X OF G TM IS WRITTEN IN THE FORM X = P(?/?XI) ?XI FOR SMOOTH FUNCTIONS XI ON EACH ??1(U) FURTHERMORE, SINCE KER ??= V , THE DIFFERENTIAL ?? IS GIVEN BY ??= P(?/?XI) ?DXI WE DE?NE A METRIC G ON THE BUNDLE G TM BY G(X, Y ) = X GIJXIY J (13) FOR EVERY SECTION X = P(?/?XI) ?XI AND Y = P(?/?XJ) ?Y J WE ALSO SET CIJK = ?GIJ ?YK = ?YI ?YJ ?YK THEN WE DE?NE A SYMMETRIC TENSOR ?ELD C : ?3 G TM ?R BY C(X, Y, Z) = X CIJKXIY JZK (14) FOR ALL SECTIONS X, Y , Z OF G TM IT IS TRIVIAL THAT C VANISHES IDENTICALLY IF AND ONLY IF G IS A RIEMANNIAN METRIC ON M THIS TENSOR ?ELD C IS CALLED THE CARTAN TENSOR ?ELD IN THE SEQUEL, WE USE THE NOTATION A0( G TM) FOR THE SPACE OF SMOOTH SECTIONS OF G TM SINCE G TM IS NATURALLY IDENTI?ED WITH V ?= KER ??, ANY SECTION X OF G TM IS CONSIDERED AS A SECTION OF V WE DENOTE BY XV THE SECTION OF V CORRESPONDING TO X ?A0( G TM): A0( G TM) ?X = X ? ?XI ?XI ?? X ? ?YI ?XI := XV ?A0(V ) THE FOLLOWING IS TRIVIAL SINCE (12) IS EXACT ??(XV ) = (15) FOR EVERY X ?A0( G TM) THE MULTIPLIER GROUP R+ ?= {CI ?GL(N, R); C ?R+} ?GL(N, R) ACTS ON THE TOTAL SPACE BY MULTIPLICATION M? : TM × ?V = (X, Y) ??V = (X, ?Y) ?TM × FOR EVERY ? ?R+ THIS ACTION INDUCES A CANONICAL SECTION E OF V DE?NED BY E(V) = (V, V) FOR ALL V ?TM × BY THE HOMOGENEITY OF F, WE HAVE E(F) = D DT T=0F(X, Y + TE) = F WE SHALL CONSIDER E AS A SECTION OF G TM, AND WE DENOTE IT BY THE SAME NOTATION E, THAT IS, E(X, Y) = P(?/?XI) ?YI THIS SECTION E IS CALLED THE TAUTOLOGICAL SECTION OF G TM THEN IT IS EASILY SHOWN THAT F = P G(E, E) AND C(E, •, •) ?0 (16) HANDBOOK OF GLOBAL ANALYSIS