THE GEOMETRY OF CURVATURE HOMOGENEOUS PSEUDO-RIEMANNIAN MANIFOLDS

By ANTHONY JOHNSON

CONTENTS 1 THE GEOMETRY OF THE RIEMANN CURVATURE TENSOR 1 1 INTRODUCTION 1 2 BASIC GEOMETRICAL NOTIONS VECTOR SPACES WITH SYMMETRIC INNER PRODUCTS VECTOR BUNDLES, CONNECTIONS, AND CURVATURE HOLONOMY AND PARALLEL TRANSLATION A?NE MANIFOLDS, GEODESICS, AND COMPLETENESS PSEUDO-RIEMANNIAN MANIFOLDS SCALAR WEYL INVARIANTS 1 3 ALGEBRAIC CURVATURE TENSORS AND HOMOGENEITY ALGEBRAIC CURVATURE TENSORS CANONICAL CURVATURE TENSORS THE WEYL CONFORMAL CURVATURE TENSOR MODELS VARIOUS NOTIONS OF HOMOGENEITY KILLING VECTOR ?ELDS NILPOTENT CURVATURE 1 4 CURVATURE HOMOGENEITY – A BRIEF LITERATURE SURVEY SCALAR WEYL INVARIANTS IN THE RIEMANNIAN SETTING RELATING CURVATURE HOMOGENEITY AND HOMOGENEITY MANIFOLDS MODELED ON SYMMETRIC SPACES HISTORICAL SURVEY 1 5 RESULTS FROM LINEAR ALGEBRA SYMMETRIC AND ANTI-SYMMETRIC OPERATORS THE SPECTRUM OF AN OPERATOR JORDAN NORMAL FORM SELF-ADJOINT MAPS IN THE HIGHER SIGNATURE SETTING TECHNICAL RESULTS CONCERNING DI?ERENTIAL EQUATIONS 1 6 RESULTS FROM DI?ERENTIAL GEOMETRY PRINCIPLE BUNDLES GEOMETRIC REALIZABILITY THE CANONICAL ALGEBRAIC CURVATURE TENSORS COMPLEX GEOMETRY RANK 1-SYMMETRIC SPACES CONFORMAL COMPLEX SPACE FORMS K¨AHLER GEOMETRY 1 7 THE GEOMETRY OF THE JACOBI OPERATOR THE JACOBI OPERATOR THE HIGHER ORDER JACOBI OPERATOR THE CONFORMAL JACOBI OPERATOR THE COMPLEX JACOBI OPERATOR 1 8 THE GEOMETRY OF THE CURVATURE OPERATOR THE SKEW-SYMMETRIC CURVATURE OPERATOR THE CONFORMAL SKEW-SYMMETRIC CURVATURE OPERATOR THE STANILOV OPERATOR THE COMPLEX SKEW-SYMMETRIC CURVATURE OPERATOR THE SZAB´O OPERATOR 1 9 SPECTRAL GEOMETRY OF THE CURVATURE TENSOR ANALYTIC CONTINUATION DUALITY BOUNDED SPECTRUM THE CONFORMAL AND COMPLEX JACOBI OPERATORS THE STANILOV AND THE SZAB´O OPERATORS 2 CURVATURE HOMOGENEOUS GENERALIZED PLANE WAVE MANIFOLDS 2 1 INTRODUCTION 2 2 GENERALIZED PLANE WAVE MANIFOLDS THE GEODESIC STRUCTURE THE CURVATURE TENSOR THE GEOMETRY OF THE CURVATURE TENSOR LOCAL SCALAR INVARIANTS PARALLEL VECTOR ?ELDS AND HOLONOMY JACOBI VECTOR ?ELDS ISOMETRIES SYMMETRIC SPACES 2 3 MANIFOLDS OF SIGNATURE (2, 2) IMMERSIONS AS HYPERSURFACES IN ?AT SPACE SPECTRAL PROPERTIES OF THE CURVATURE TENSOR A COMPLETE SYSTEM OF INVARIANTS ESTIMATING KP,Q IF MIN(P, Q) = 2 2 4 MANIFOLDS OF SIGNATURE (2, 4) 2 5 PLANE WAVE HYPERSURFACES OF NEUTRAL SIGNATURE (P, P) CURVATURE HOMOGENEITY 2 6 PLANE WAVE MANIFOLDS WITH FLAT FACTORS 2 7 NIK?CEVI´C MANIFOLDS LOCAL ISOMETRY INVARIANTS THE SPECTRAL GEOMETRY OF THE CURVATURE TENSOR 2 8 DUNN MANIFOLDS MODELS AND THE STRUCTURE GROUPS INVARIANTS WHICH ARE NOT OF WEYL TYPE 2 9 K-CURVATURE HOMOGENEOUS MANIFOLDS I A?NE INVARIANTS CHANGING THE SIGNATURE INDECOMPOSABILITY 2 10 K-CURVATURE HOMOGENEOUS MANIFOLDS II 2 10 1 MODELS 2 10 2 ISOMETRY GROUPS 3 OTHER PSEUDO-RIEMANNIAN MANIFOLDS 3 1 INTRODUCTION 3 2 LORENTZ MANIFOLDS GEODESICS AND CURVATURE RICCI BLOWUP 3 3 SIGNATURE (2, 2) WALKER MANIFOLDS OSSERMAN CURVATURE TENSORS OF SIGNATURE (2, 2) INDE?NITE K¨AHLER OSSERMAN MANIFOLDS JORDAN OSSERMAN MANIFOLDS WHICH ARE NOT NILPOTENT CONFORMALLY OSSERMAN MANIFOLDS 3 4 GEODESIC COMPLETENESS AND RICCI BLOWUP THE GEODESIC EQUATION JORDAN OSSERMAN WALKER MANIFOLDS 3 5 FIEDLER MANIFOLDS GEOMETRIC PROPERTIES OF FIEDLER MANIFOLDS FIEDLER MANIFOLDS OF SIGNATURE (2, 2) NILPOTENT JACOBI MANIFOLDS OF ORDER 2R NILPOTENT JACOBI MANIFOLDS OF ORDER 2R + 1 SZAB´O NILPOTENT MANIFOLDS OF ARBITRARILY HIGH ORDER 4 THE CURVATURE TENSOR 4 1 INTRODUCTION 4 2 TOPOLOGICAL RESULTS REAL VECTOR BUNDLES BUNDLES OVER PROJECTIVE SPACES CLI?ORD ALGEBRAS IN ARBITRARY SIGNATURES RIEMANNIAN CLI?ORD ALGEBRAS VECTOR ?ELDS ON SPHERES METRICS OF HIGHER SIGNATURES ON SPHERES EQUIVARIANT VECTOR ?ELDS ON SPHERES GEOMETRICALLY SYMMETRIC VECTOR BUNDLES 4 3 GENERATORS FOR THE SPACES ALG0 AND ALG1 A LOWER BOUND FOR ?(M) AND FOR ?1(M) 4 4 JORDAN OSSERMAN ALGEBRAIC CURVATURE TENSORS NEUTRAL SIGNATURE JORDAN OSSERMAN TENSORS RIGIDITY RESULTS FOR JORDAN OSSERMAN TENSORS 4 5 THE SZAB´O OPERATOR SZAB´O 1-MODELS BALANCED SZAB´O PSEUDO-RIEMANNIAN MANIFOLDS 4 6 CONFORMAL GEOMETRY THE WEYL MODEL CONFORMALLY JORDAN OSSERMAN 0-MODELS CONFORMALLY OSSERMAN 4-DIMENSIONAL MANIFOLDS CONFORMALLY JORDAN IVANOV–PETROVA 0-MODELS 4 7 STANILOV MODELS 4 8 COMPLEX GEOMETRY 5 COMPLEX OSSERMAN ALGEBRAIC CURVATURE TENSORS 5 1 INTRODUCTION CLI?ORD FAMILIES COMPLEX OSSERMAN TENSORS CLASSI?CATION RESULTS IN THE ALGEBRAIC SETTING GEOMETRIC EXAMPLES CHAPTER OUTLINE 5 2 TECHNICAL PRELIMINARIES CRITERIA FOR COMPLEX OSSERMAN MODELS CONTROLLING THE EIGENVALUE STRUCTURE EXAMPLES OF COMPLEX OSSERMAN 0-MODELS REPARAMETRIZATION OF A CLI?ORD FAMILY THE DUAL CLI?ORD FAMILY COMPATIBLE COMPLEX MODELS GIVEN BY CLI?ORD FAMILIES LINEARLY INDEPENDENT ENDOMORPHISMS TECHNICAL RESULTS CONCERNING CLI?ORD ALGEBRAS 5 3 CLI?ORD FAMILIES OF RANK 5 4 CLI?ORD FAMILIES OF RANK THE TENSOR C0A? , ?+ C1AJ1 + C2AJ2 5 5 CLI?ORD FAMILIES OF RANK TECHNICAL RESULTS THE TENSOR A = C1AJ1 + C2AJ2 + C3AJ3 5 7 TENSORS A = C0A? , ?+ C1AJ1 + + C?AJ?FOR ??4 6 STANILOV–TSANKOV THEORY 6 1 INTRODUCTION JACOBI TSANKOV MANIFOLDS SKEW TSANKOV MANIFOLDS STANILOV–VIDEV MANIFOLDS JACOBI VIDEV MANIFOLDS AND 0-MODELS 6 2 RIEMANNIAN JACOBI TSANKOV MANIFOLDS AND 0-MODELS RIEMANNIAN JACOBI TSANKOV 0-MODELS RIEMANNIAN ORTHOGONALLY JACOBI TSANKOV 0-MODELS RIEMANNIAN JACOBI TSANKOV MANIFOLDS 6 3 PSEUDO-RIEMANNIAN JACOBI TSANKOV 0-MODELS JACOBI TSANKOV 0-MODELS NON JACOBI TSANKOV 0-MODELS WITH J X = 0 ?X 0-MODELS WITH JX JY = 0 ?X, Y ?V 0-MODELS WITH AXYAZW = 0 ?X, Y, Z, W ?V 6 4 A JACOBI TSANKOV 0-MODEL WITH JX JY ?= 0 FOR SOME X, Y THE MODEL M14 ISOMETRY INVARIANTS 6 5 RIEMANNIAN SKEW TSANKOV MODELS AND MANIFOLDS RIEMANNIAN SKEW TSANKOV MODELS 3-DIMENSIONAL SKEW TSANKOV MANIFOLDS IRREDUCIBLE 4-DIMENSIONAL SKEW TSANKOV MANIFOLDS FLATS IN A RIEMANNIAN SKEW TSANKOV MANIFOLD 6 6 JACOBI VIDEV MODELS AND MANIFOLDS EQUIVALENT PROPERTIES CHARACTERIZING JACOBI VIDEV MODELS DECOMPOSING JACOBI VIDEV MODELS CHAPTER IS DUE TO TRICERRI AND VANHECKE (1986) IN THE RIEMANNIAN SETTING AND TO CAHEN, LEROY, PARKER, TRICERRI, AND VANHECKE (1991) IN THE LORENTZIAN SET- TING AND DEALS WITH MANIFOLDS WHICH ARE 0-CURVATURE MODELED ON IRREDUCIBLE SYMMETRIC SPACES WE CONCLUDE SECTION 1 4 WITH A SURVEY OF THE LITERATURE