TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY, COMPLEX ANALYSIS AND MATHEMATICAL PHYSICS

By JOSEPH TAYLOR

CONTENTS MODULI SPACE OF KILLING HELICES OF LOW ORDERS ON A COMPLEX SPACE FORM TYPE TWO KILLING HELICES OF PROPER ORDER FOUR ON A COMPLEX PROJECTIVE PLANE REAL ANALYTICITY OF THE HYPERKAHLER ALMOST K¨AHLER MANIFOLDS KCC AND LINEAR STABILITY FOR THE PARKINSON TREMOR MODEL RIGHT-INVARIANT METRIC FERMI-WALKER PARALLEL TRANSPORT, TIME EVOLUTION OF A SPACE CURVE AND THE SCHR¨ODINGER EQUATION AS A MOVING CURVE REMARKS ON BICOMPLEX VARIABLES AND OTHER SIMILAR VARIABLES INTEGRABILITY, CURVATURE AND DESCRIPTION OF PHOTON-LIKE OBJECTS COMPLEX SUBMANIFOLDS AND LAGRANGIAN SUBMANIFOLDS ASSO- CIATE WITH MINIMAL SURFACES IN TORI ABOUT RESONANCES FOR SCHR¨ODINGER OPERATORS WITH SHORT RANGE SINGULAR PERTURBATION SOLITON EQUATIONS WITH DEEP REDUCTIONS GENERALIZED FOURIER TRANSFORMS ON SOME DEFORMATIONS OF ALMOST COMPLEX STRUCTURES OF 6-DIMENSIONAL SUBMANIFOLDS IN THE OCTONIONS ON THE REAL HYPERSURFACES OF LOCALLY CONFORMALLY-K¨AHLER MANIFOLDS NORMAL FRAMES AND LINEAR TRANSPORTS ALONG PATHS IN LINE BUN- DLES APPLICATIONS TO CLASSICAL ELECTRODYNAMICS EXISTENCE AND UNIQUENESS RESULTS FOR THE SCHR¨ODINGER — POISSON SYSTEM BELOW THE ENERGY NORM DOUBLE-COMPLEX QUADRATIC FORMS E?ECTS OF ?NITE ARITHMETIC IN NUMERICAL COMPUTATIONS EXACT SOLUTIONS OF THE MANAKOV SYSTEM THE MANAKOV SYSTEM AS TWO MOVING INTERACTING CURVES CLUSTER SETS AND PERIODICITY IN SOME STRUCTURE FRACTALS SECTIONAL CURVATURES OF SOME HOMOGENEOUS REAL HYPERSURFACES IN A COMPLEX PROJECTIVE SPACE ON THREE-PARAMETRIC LIE GROUPS AS QUASI-K¨AHLER MANIFOLDS WITH KILLING NORDEN METRIC APPLICATION OF UNSCENTED AND EXTENDED KALMAN ?LTERING FOR ESTIMATING QUATERNION MOTION COUNTEREXAMPLES OF COMPACT TYPE TO THE GOLDBERG CONJECTURE AND VARIOUS VERSION OF THE CONJECTURE DISPERSION AND ASYMPTOTIC PRO?LES FOR KIRCHHO?EQUATIONS GEOMETRY OF STATISTICAL MANIFOLDS AND ITS GENERALIZATION ON SOME NON-INTEGRABLE ALMOST CONTACT MANIFOLDS WITH NOR- DEN METRIC OF DIMENSION COMPLEXI?CATION OF THE PROPAGATOR FOR THE HARMONIC OSCILLATOR SOME CRITICAL ALMOST K¨AHLER STRUCTURES WITH A ?XED K¨AHLER CLASS ON THE HYPOELLIPTICITY OF SOME CLASSES OF OVERDETERMINED SYSTEMS OF DI?ERENTIAL AND PSEUDODI?ERENTIAL OPERATORS LIE GROUPS AS FOUR-DIMENSIONAL RIEMANNIAN PRODUCT MANIFOLDS DI?ERENTIAL FORMS OF MANY DOUBLE-COMPLEX VARIABLES ISOMETRIC IMMERSIONS AND EXTRINSIC SHAPES OF SMOOTH CURVES LIE GROUPS AS FOUR-DIMENSIONAL CONFORMAL K¨AHLER MANIFOLDS WITH NORDEN METRIC B¨ACKLUND TRANSFORMATIONS AND RIEMANN-HILBERT PROBLEM FOR N WAVE EQUATIONS WITH ADDITIONAL SYMMETRIES MODULI SPACE OF KILLING HELICES OF LOW ORDERS ON A COMPLEX SPACE FORM WE GIVE A REPORT ON THE MODULI SPACE OF HELICES OF PROPER ORDER LESS THAN ARE NULL VECTOR ?ELDS ALONG ? THESE CONSTANTS ?1, , ?D?1 ARE CALLED THE GEODESIC CURVATURES OF ? AND THE SYSTEM YI THE FRENET FRAME OF ? WE CALL A HELIX KILLING IF IT IS GENERATED BY SOME KILLING VECTOR ?ELD ON M ON REAL SPACE FORMS, WHICH ARE STANDARD SPHERES, EUCLIDEAN SPACES AND REAL HYPERBOLIC SPACES, ALL HELICES ARE KILLING AND LENGTHS OF CLOSED HELICES ARE GIVEN BY THEIR GEODESIC CURVATURES BUT ON A COMPLEX SPACE FORM THE SITUATION IS DI?ERENT WE STUDY THE DI?ERENCE ON LAMINATIONS ON THE MODULI SPACES OF KILLING HELICES WHICH ARE INDUCED BY THE LENGTH SPECTRUM , WHERE [??1,?2] DENOTES THE CONGRUENCE CLASS OF HELICES OF PROPER ORDER 3 ON SN(C) WITH GEODESIC CURVATURES ?1, ?2 (SEE FIGURE 1) TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY RM N(C) C SN(C) [BOUNDED] [UNBOUNDED] C AND WE SHALL DENOTE BY [??,?] THE CONGRUENCE CLASS OF CIRCLES WITH GEODESIC CURVATURE ? AND COMPLEX TORSION ?12 = ? ON A COMPLEX SPACE FORM CM N(C) OF CONSTANT HOLOMORPHIC SECTIONAL CURVATURE C TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY CP N(C) DE?NED BY Fµ = ? ? ? ? ? ? ? [??,0] ? > , IF µ = 0, [??,?] ? 3C??(4?2 + C)?3/2 = µ , IF 0 < µ < 1, [??,1] ? > CHN(C) DE?NED BY Fµ = ? ? ? ? ? ? ? ? ? ? ? [??,0] ? > ? 3|C|??|4?2 + C|?3/2 = µ , IF 0 < µ < ?, ?? |C|/2,? 0 < ? < , IF µ = ?, [??,1] ? > ? 3 |C|?), IF |C|/2 < ? < |C|, 1, IF ? ? |C| IN VIEW OF THE FEATURES OF THESE LAMINATIONS ON THE MODULI SPACES OF CIRCLES, WE ?ND THE SET [??,1] ? > OF CONGRUENCE CLASSES OF TRAJECTORIES TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY CP N(C) C C CM N IS BIJECTIVE TO THE SET (0, ?)2 × [0, 1] WHEN WE CONSIDER KILLING HELICES OF PROPER ORDER 4, THE RELATIONS (5 1) TURN TO ?1?23 + ?3?14 = ?2?12, ?3?23 + ?1?14 = ?2?34 AND ?13 = ?24 = 0 CONSIDERING THE INITIAL FRAME WE ?ND A HELIX OF PROPER ORDER 4 ON CM N (N ?2) IS ESSENTIAL KILLING IF AND ONLY IF ITS COMPLEX TORSIONS SATISFY ONE OF THE FOLLOWING CONDITIONS: TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY CP N(C) ?(0, ?] IS SMOOTH ON EACH LEAF Gµ WITH RESPECT TO THE CANONICAL INDUCED EUCLIDEAN DI?ERENTIAL STRUCTURE EACH LEAF IS MAXIMAL WITH RESPECT TO THIS PROPERTY 1) THE LEAF G0 CONSISTS OF CONGRUENCE CLASSES OF CLOSED HELICES OF PROPER ORDER 3 SATISFYING L([?(?1, ?2)]) = 2 + 3C 2) THE LEAF Gµ (µ ?= 0) CONSISTS OF CLOSED HELICES OF PROPER ORDER 3 IF AND ONLY IF µ = ±Q(9P2?Q2)(3P2+Q2)?3/2 WITH SOME RELATIVELY PRIME POSITIVE INTEGERS P, Q SATISFYING P > Q TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY CP N WHICH PRESERVES THE FOLIATION STRUCTURE THE READER SHOULD COMPARE FIGURES 1 AND 5 HE CAN ?ND DI?ERENCE BETWEEN THEIR FEATURES NEAR THE HALF LINE (?, 0) ? > CP N C C CP N(C) = [?(?1, ?2, ??1)] ?1, ?2 > 2 + 3C)3/2 < ? ? ? ? ? TOPICS IN CONTEMPORARY DIFFERENTIAL GEOMETRY CHN(C) WE SHOULD NOTE THAT SOME ESSENTIAL KILLING HELICES OF PROPER ORDER 4 ARE OBTAINED AS TRAJECTORIES FOR CANONICAL MAGNETIC ?ELDS ON REAL HYPERSURFACES OF TYPE A1 IN CM N (SEE [3]) [BOUNDED] [UNBOUNDED] C C C