zoll manifolds and complex surfaces we classify compact surfaces with torsion free ane connec tions for which every geodesic is a simple closed curve in the process we obtain completely new proofs of all the major results 4 concerning the riemannian case in contrast to previous work our approach is twistor theoretic and depends fundamentally on the fact that up to biholomorphism there is only one complex structure on cp2

kahlerian twistor spaces the work of r penrose has shown the deep relationship between the conformal structure of minkowski space and the complex geometry of lines in projective three space this transformation applies to a more general class of conformal structures and in particular there is a riemannian version described in 5 which is itself of interest to mathematical physicists basically to each oriented riemannian 4manifold x one associates canonically a 6 manifold z the twistor space of x together with an almost complex structure if this structure is integrable then z becomes a complex 3 manifold with a distinguished family of projective lines on it whose geometry describes the conformal geometry of the manifold x it is natural to ask what sort of complex manifold one obtains by this process in this paper we ask the more specific question which compact 4c manifolds have ka hlerian twistor spaces 1 the answer rather surprisingly is that there are just two examples the 4 sphere s and the complex projective plane p2 c each with their standard homogeneous conformal structures the twistor space of s 4 is the 3 dimensional projective space p3 c and that of p 2 q is the manifold f2 c of flags in c 3 the essential point of the proof of this result is the remark that if z is a twistor space with a kahler metric then the canonical bundle k is negative using vanishing theorems and the riemann roch theorem we then narrow down the possibilities for z to four cases projective space the flag manifold the intersection of two quadrics in p5 and the double covering of p3 branched over a non singular quartic surface the last two candidates cannot be realized however because the value of their euler characteristic is incompatible with their being twistor spaces throughout the proof we make essential use of a real structure on z which reduces considerably the occurrence of singularities at various stages aside from this our methods are the standard ones of algebraic geometry the four examples to which the proof leads us are all fano threefolds if we drop the requirement of compactness on x then it is quite possible for an open set of a fano threefold to be a twistor space in particular we show that the family of conies lying on the intersection of two

toric anti self dual einstein metrics via complex geometry using the twistor correspondence we give a classification of toric anti self dual einstein metrics each such metric is essentially determined by an odd holomorphic function this explains how the einstein metrics fit into the classification of general toric anti self dual metrics given in an earlier paper donaldson and fine in math ann 336 2 281 309 2006 the results complement the work of calderbank pedersen j differential geom 60 3 485 521 2002 who describe where the einstein metrics appear amongst the joyce spaces leading to a different classification taking the twistor transform of our result gives a new proof of their theorem

the complex monge ampere equation zoll metrics and algebraization let m be a real analytic riemannian manifold an adapted complex structure on tm is a complex structure on a neighborhood of the zero section such that the leaves of the riemann foliation are complex submanifolds this structure is called entire if it may be extended to the whole of tm we prove here that the only real analytic zoll metric on the n sphere with an entire adapted complex structure on tm is the round sphere using similar ideas we answer a special case of an algebraization question raised by the first author characterizing some stein manifolds as affine algebraic in terms of plurisubharmonic exhaustion functions satisfying the homogeneous complex monge ampere equation the result presented here is an extension to higher dimensions of an observation attributed to w stoll for the case of riemann surfaces

degree of mobility for metrics of lorentzian signature and parallel 0 2 tensor fields on cone manifolds degree of mobility of a pseudo riemannian metric is the dimension of the space of metrics geodesically equivalent to it we describe all possible values of the degree of mobility on a simply connected n dimensional manifold of lorentzian signature as an application we calculate all possible differences between the dimension of the projective and the isometry groups one of the main new technical results in the proof is the description of all parallel symmetric 0 2 tensor fields on cone manifolds of signature n 1 2

anti self dual four manifolds with a parallel real spinor antiselfdual metrics in the signature that admit a covariantly constant real spinor are studied it is shown that finding such metrics reduces to solving a fourthorder integrable partial differ

two dimensional reductions of the self dual yang mills equations in self dual spaces many completely integrable equations have been shown to be reductions of the yang mills equations in a flat space in this paper two dimensional symmetry reductions of the self dual yang mills equations together with the corresponding linear equations in an arbitrary self dual space are classified the reduced equations are given in different forms some new equations are found in particular one which generalizes the sine gordon equation and the liouville equation

algebraic dimension of twistor spaces si l espace twisteur d une variete compacte autoduale est moishezon la classe conforme autoduale contient une metrique de courbure scalaire positive

on twistor spaces of anti self dual hermitian surfaces we consider a complex surface m with anti self dual hermitian metric h and study the holomorphic properties of its twistor space z we show that the naturally defined divisor line bundle x is isomorphic to the 1 2 power of the canonical bundle of z if and only if there is a kahler metric of zero scalar curvature in the conformal class of h this has strong consequences on the geometry of m which were also found by c boyer 3 using completely different methods we also prove the existence of a very close relation between holomorphic vector fields on m and z in the case that m is compact and kahler

the numerically stable reconstruction of jacobi matrices from spectral data we present an expose of the elementary theory of jacobi matrices and in particular their reconstruction from the gaussian weights and abscissas many recent works propose use of the diagonal hermitian lanczos process for this purpose we show that this process is numerically unstable we recall rutishauser s elegant and stable algorithm of 1963 based on plane rotations implement it efficiently and discuss our numerical experience we also apply rutishauser s algorithm to reconstruct a persymmetric jacobi matrix from its spectrum in an efficient and stable manner

a solution of a problem of sophus lie normal forms of two dimensional metrics admitting two projective vector fields we give a complete list of normal forms for the two dimensional metrics that admit a transitive lie pseudogroup of geodesic preserving transformations and we show that these normal forms are mutually non isometric this solves a problem posed by sophus lie

singular hyperbolic monopoles we study the twistor theory of singular hyperbolic su 2 monopoles following the approach taken by kronheimer 9 in the euclidean case we use our results to show that the moduli space of charge 1 monopoles possesses a natural 2 sphere of scalar flat kahler metrics in the zero mass limit the metrics reduce to a class of metrics first studied by lebrun in 10

two dimensional metrics admitting precisely one projective vector field we give a complete list of 2 dimensional metrics that admit an essential projective vector field this solves a problem explicitly posed by sophus lie in 1882

picard and chazy solutions to the painleve vi equation we study the solutions of a particular family of painleve vi equations with parameters beta gamma 0 delta frac 1 2 and 2 alpha 2 mu 1 2 for 2 mu in mathbb z we show that in the case of half integer mu all solutions can be written in terms of known functions and they are of two types a two parameter family of solutions found by picard and a new one parameter family of classical solutions which we call chazy solutions we give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0 1 infty and their nonlinear monodromy we study the structure of analytic continuation of the solutions to the pvi mu equation for any mu such that 2 mu in mathbb z as an application we classify all the algebraic solutions for mu half integer we show that they are in one to one correspondence with regular polygons or star polygons in the plane for mu integer we show that all algebraic solutions belong to a one parameter family of rational solutions

matrices with prescribed eigenvalues and principal submatrices abstract we give a necessary and sufficient condition for the existence of a square matrix with prescribed eigenvalues and prescribed complementary principal blocks

on a new class of holonomy groups in pseudo riemannian geometry holonomy groups were introduced by elie cartan in the twenties 15 16 for the study of riemannian symmetric spaces and since then the classi cationof holonomy groups has remained one of the classical problems in di erentialgeometry de nition 1 let m be a smooth manifold endowed with an a ne symmetricconnection the holonomy group of is a subgroup hol gl t

metrisability of two dimensional projective structures we carry out the programme of r liouville 19 to construct an explicit local obstruction to the existence of a levi civita connection within a given projective structure on a surface the obstruction is of order 5 in the components of a connection in a projective class it can be expressed as a point invariant for a second order ode whose integral curves are the geodesics of or as a weighted scalar projective invariant of the projective class if the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case in the generic case they are expressed by the vanishing of two invariants of order 6 in the connection in degenerate cases the sufficient obstruction isof order at most 8

metrisability of painleve equations we solve the metrisability problem for the six painleve equations and more generally for all 2nd order ordinary differential equations with the painleve property and determine for which of these equations their integral curves are geodesics of a pseudo riemannian metric on a surface

representations and routing for cayley graphs computer networks it is shown that all finite cayley graphs can be represented by generalized chordal rings gcr an example borel cayley graph is used to illustrate the generation of gcr representations a sufficient condition is given for the representation of a cayley graph as a chordal ring cr with the integer labeling of gcr representations a straightforward progressive routing algorithm based on table look up is summarized

splitting and gluing lemmas for geodesically equivalent pseudo riemannian metrics two metrics g and are geodesically equivalent if they share the same unparameterized geodesics we introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics such as the classification of local normal forms and the lie problem the description of projective vector fields to the case when the 1 1 tensor g g gkj has one real eigenvalue or two complex conjugate eigenvalues and give first applications as a part of the proof of the main result we generalise the topalov sinjukov hierarchy theorem for pseudo riemannian metrics 2011 american mathematical society

an example of an h space the good cut equation for a specific asymptotic shear is solved and the metric of the associated h space is found to be type n asymptotically flat and positive frequency

a criterion for compatibility of conformal and projective structures in a space time m a conformal structure is defined by the distribution of light cones geodesics are traced by freely falling particles and the collection of all unparameterized geodesics determines the projective structure of m the article contains a formulation of the necessary and sufficient conditions for these structures to be compatible i e to come from a metric tensor which is then unique up to a constant factor the theorem applies to all dimensions and signatures

proper conformal symmetries in self dual einstein spaces proper conformal symmetries in self dual einstein spaces are considered it is shown that such symmetries are admitted only by the einstein spaces of the type n n spaces of the type n are considered in details existence of the proper conformal killing vector implies existence of the isometric covariantly constant and null killing vector it is shown that there are two classes of n metrics admitting proper conformal symmetry they can be distinguished by analysis of the associated anti self dual asd null strings both classes are analyzed in details the problem is reduced to single linear partial differential equation pde some general and special solutions of this pde are presented

toric anti self dual 4 manifolds via complex geometry using the twistor correspondence this article gives a one to one correspondence between germs of toric anti self dual conformal classes and certain holomorphic data determined by the induced action on twistor space recovering the metric from the holomorphic data leads to the classical problem of prescribing the cech coboundary of 0 cochains on an elliptic curve covered by two annuli the classes admitting kahler representatives are described each such class contains a circle of kahler metrics this gives new local examples of scalar flat kahler surfaces and generalises work of joyce duke math j 77 3 519 552 1995 who considered the case where the distribution orthogonal to the torus action is integrable

matrices with prescribed rows and invariant factors abstract in this paper we solve the problem of the existence of an n n matrix over an arbitrary field when its invariant polynomials and either some rows or columns are prescribed the solution is given in terms of invariant factor inequalities and of majorization inequalities involving controllability indices and the degrees of the invariant polynomials

1 1 dimensional separation of variables in this paper we explore general conditions which guarantee that the geodesic flow on a two dimensional manifold with indefinite signature is locally separable this is equivalent to showing that a two dimensional natural hamiltonian system on the hyperbolic plane possesses a second integral of motion which is a quadratic polynomial in the momenta associated with a secind rank killing tensor we examine the possibility that the integral is preserved by the hamiltonian flow on a given energy hypersurface only weak integrability and derive the additional requirement necessary to have conservation at arbitrary values of the hamiltonian strong integrability using null coordinates we show that the leading order coefficients of the invariant are arbitrary functions of one variable in the case of weak integrability these functions are quadratic polynomials in the coordinates in the case of strong integrability we show that for 1 1 dimensional systems there are three possible types of conformal killing

geodesic rigidity of conformal connections on surfaces we show that a conformal connection on a closed oriented surface sigma of negative euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics as a corollary it follows that the unparametrised geodesics of a riemannian metric on sigma determine the metric up to constant rescaling it is also shown that every conformal connection on the 2 sphere lies in a complex 5 manifold of conformal connections all of which share the same unparametrised geodesics

explicit construction of hyperdominant symmetric matrices with assigned spectrum abstract we prove by explicit construction the existence of at least one nontrivial symmetric hyperdominant matrix with assigned nonnegative spectrum in addition we construct symmetric matrices with assigned spectra and several specific nonhyperdominant types of sign patterns

twistor geometry of a pair of second order odes we discuss the twistor correspondence between path geometries in three dimensions with vanishing wilczynski invariants and anti self dual conformal structures of signature 2 2 we show how to reconstruct a system of odes with vanishing invariants for a given conformal structure highlighting the ricci flat case in particular using this framework we give a new derivation of the wilczynski invariants for a system of odes whose solution space is endowed with a conformal structure we explain how to reconstruct the conformal structure directly from the integral curves and present new examples of systems of odes with point symmetry algebra of dimension four and greater which give rise to anti self dual structures with conformal symmetry algebra of the same dimension some of these examples are 2 2 analogues of plane wave space times in general relativity finally we discuss a variational principle for twistor curves arising from the finsler structures with scalar flag curvature

interconnection networks based on a generalization of cube connected cycles a generalization of the cube connected cycles of preparata and vuillemin is described which retains the symmetry of these architectures while allowing for constructions of greater density and of arbitrary degree these constructions are of a type known as cayley graphs and their analysis is greatly facilitated by the applicability of methods from abstract algebra

twistors kahler manifolds and bimeromorphic geometry i by considering deformations of the moishezon twistor spaces of cp2 cp2 constructed in 20 we showthat theblowupof c2 at n points in general position admits an asymptotically flat scalar flat kahler metric in each kahler class at least provided that the given points are nearly collinear department of mathematics suny at stony brook stony brook new york 11794 e mail address claude math sunysb edu this content downloaded from 157 55 39 153 on mon 19 sep 2016 04 58 00 utc all use subject to http about jstor org terms

nonlinear gravitons null geodesics and holomorphic disks we develop a global twistor correspondence for pseudo riemannian conformal structures of signature with self dual weyl curvature near the conformal class of the standard indefinite product metric on s 2 s 2 there is an infinitedimensional moduli space of such conformal structures and each of these has the surprising global property that its null geodesics are all periodic each such conformal structure arises from a family of holomorphic disks in cp3 with boundary on some totally real embedding of rp 3 into cp3 an interesting sub class of these

class congruence property and two phase routing of borel cayley graphs dense symmetric graphs are useful interconnection models for multicomputer systems borel cayley graphs the densest degree 4 graphs for a range of diameters are attractive candidates however the group theoretic representation of these graphs makes the development of efficient routing algorithms difficult in earlier reports we showed that all degree 4 borel cayley graphs have generalized chordal ring gcr and chordal ring cr representations we present the class congruence property and use this property to develop the two phase routing algorithm for borel cayley graphs in a special gcr representation the algorithm requires a small space complexity of o p k for n p spl times k nodes although suboptimal the algorithm finds paths with length bounded by 2d where d is the diameter furthermore our computer implementation of the algorithm on networks with 1 081 and 15 657 nodes shows that the average path length is on the order of the diameter the performance of the algorithm is compared with that of existing optimal and suboptimal algorithms

neutral calabi yau structures on kodaira manifolds we construct neutral calabi yau metrics and hypersymplectic structures on some kodaira manifolds our structures are symmetric with respect to the central tori

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

global anti self dual yang mills fields in split signature and their scattering this article concerns solutions to the anti self dual yang mills asdym equations in split signature that are global on the double cover of the appropriate conformally compactified minkowski space e m s 2 s 2 ward s asdym twistor construction is adapted to this geometry using a correspondence between points of e m and holomorphic discs in cp 3 twistor space with boundary on the real slice rp 3 smooth global u n solutions to the asdym equations on e m are shown to be in 1 1 correspondence with pairs consisting of an arbitrary holomorphic vector bundle e over cp 3 together with a positive definite hermitian metric h on e rp 3 there are no topological or other restrictions on the bundle e in ultrahyperbolic signature solutions are generically non analytic or only finitely differentiable and such solutions arise from a corresponding choice of regularity for h when e is trivial the twistor data consists of the hermitian matrix function h on rp 3 up to constants and the correspondence provides a nonlinear generalisation of the x ray transform in general it provides a higher dimensional analogue of the inverse scattering transform in which h plays the role of the reflection coefficient ande the algebraic data explicit examples are constructed for different choices of the topology of e a scattering problem for asdym fields on affine minkowski spacein split signature is set up and it is shown that sufficiently small data at past null infinity uniquely determines data at future null infinity by taking a family of holonomies associated to the initial data followed by a sequence of two birkhoff factorizations the scattering map is simple at the level of the holonomies but non trivial at the level of the connection in the non abelian case

anti self dual conformal structures with null killing vectors from projective structures using twistor methods we explicitly construct all local forms of four dimensional real analytic neutral signature anti self dual conformal structures m g with a null conformal killing vector we show that m is foliated by anti self dual null surfaces and the two dimensional leaf space inherits a natural projective structure the twistor space of this projective structure is the quotient of the twistor space of m g by the group action induced by the conformal killing vector

the local equivalence problem for d 2 y dx 2 f x y dy dx and the painleve transcendents on resout le probleme d equivalence locale pour d 2 y dx 2 f x y dy dx sous le groupe naturel des changements de coordonnees x x y x y il y a trois invariants de base qui s annulent si et seulement si l equation est equivalente a d 2 y dx 2 0 on montre que deux des invariants s annulent pour les 6 transcendants de painleve et le 3eme peut etre utilise pour produire un ensemble complet d invariants

projective compactifications and einstein metrics for complete affine manifolds we introduce a definition of compactification based on the projective differential geometry i e geodesic path data of the given connection the definition of projective compactness involves a real parameter called the order of projective compactness for volume preserving connections this order is captured by a notion of volume asymptotics that we define these ideas apply to complete pseudo riemannian spaces via the levicivita connection and thus provide a notion of compactification alternative to conformal compactification for each order we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order thus also providing many local examples distinguished classes of projectively compactified geometries of orders one and two are associated with ricci flat connections and non ricci flat einstein metrics respectively conversely these geometric conditions are shown to force the indicated order of projective compactness these special compactifications are shown to correspond to normal solutions of classes of natural linear pde socalled first bgg equations or equivalently holonomy reductions of projective cartan tractor connections this enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity finally we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above msc2010 primary 53a20 53b21 53b10 secondary 53c29 35n10 53a30 54d35 both authors gratefully acknowledge support from the royal society of new zealand via marsden grant 10 uoa 113 ac gratefully acknowledges support by project p23244 n13 of the fonds zur forderung der wissenschaftlichen forschung fwf and also the hospitality of the university of auckland

minimal lagrangian connections on compact surfaces abstract we introduce the notion of a minimal lagrangian connection on the tangent bundle of a manifold and classify all such connections in the case where the manifold is a compact oriented surface of non vanishing euler characteristic combining our classification with results of labourie and loftin we conclude that every properly convex projective surface arises from a unique minimal lagrangian connection

metrisability of projective surfaces and pseudo holomorphic curves we show that the metrisability of an oriented projective surface is equivalent to the existence of pseudo holomorphic curves a projective structure mathfrak p and a volume form sigma on an oriented surface m equip the total space of a certain disk bundle z rightarrow m with a pair j mathfrak p mathfrak j mathfrak p sigma of almost complex structures a conformal structure on m corresponds to a section of z rightarrow m and mathfrak p is metrisable by the metric g if and only if g m rightarrow z is a pseudo holomorphic curve with respect to j mathfrak p and mathfrak j mathfrak p da g

local normal forms for geodesically equivalent pseudo riemannian metrics this is an extended version of the submitted paper it contains all proofs with all details and also an appendix where we explain how can one construct a complex structure by a 1 1 tensor such that all its eigenvalues are not real and such that the nijenhuis torsion vanishes

twistor theory at fifty from contour integrals to twistor strings we review aspects of twistor theory its aims and achievements spanning the last five decades in the twistor approach space time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold the twistor space after giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics anti self duality equations on yang mills or conformal curvature can be encoded into twistor cohomology these twistor correspondences yield explicit examples of yang mills and gravitational instantons which we review they also underlie the twistor approach to integrability the solitonic systems arise as symmetry reductions of anti self dual asd yang mills equations and einstein weyl dispersionless systems are reductions of asd conformal equations we then review the holomorphic string theories in twistor and ambitwistor spaces and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes finally we discuss the newtonian limit of twistor theory and its possible role in penrose s proposal for a role of gravity in quantum collapse of a wave function

hyperbolic manifolds are geodesically rigid we show that if all geodesics of two non proportional metrics on a closed manifold coincide as unparameterized curves then the manifold has a finite fundamental group or admits a local product structure this implies that if the manifold admits a metric of negative sectional curvature then two metrics on the manifold have the same geodesics if and only if they are proportional

matrices with prescribed characteristic polynomial and principal blocks ii abstract in this paper we present some new sufficient conditions for the existence of a square matrix with prescribed characteristic polynomial and two complementary principal submatrices

quantum integrability of beltrami laplace operator as geodesic equivalence given two riemannian metrics on a closed connected manifold m n we construct self adjoint differential operators cal i 0 cal i 1 cal i n 1 c 2 m n to c 0 m n such that if the metrics have the same geodesics then the operators commute with the beltrami laplace operator of the first metric and pairwise commute if the operators commute and if they are linearly independent then the metrics have the same geodesics

selfdual einstein metrics with torus symmetry it is well known that any 4 dimensional hyperkahler metric with two commuting killing fields may be obtained explicitly via the gibbons hawking ansatz from a harmonic function invariant under a killing field on 3 in this paper we find all selfdual einstein metrics of nonzero scalar curvature with two commuting killing fields they are given explicitly in terms of a local eigenfunction of the laplacian on the hyperbolic plane we discuss the relation of this construction to a class of selfdual spaces found by joyce and some einstein weyl spaces found by ward and then show that certain multipole hyperbolic eigenfunctions yield explicit formulae for the quaternion kahler quotients of pm 1 by an m 2 torus studied by galicki and lawson as a consequence we are able to place the well known cohomogeneity one metrics the quaternion kahler quotients of p2 and noncompact analogues and the more recently studied selfdual einstein hermitian metrics in a unified framework and give new complete examples

conformal killing forms on riemannian manifolds conformal killing forms are a natural generalization of conformal vector fields on riemannian manifolds they are defined as sections in the kernel of a conformally invariant first order differential operator we show the existence of conformal killing forms on nearly kahler and weak g2 manifolds moreover we give a complete description of special conformal killing forms a further result is a sharp upper bound on the dimension of the space of conformal killing forms

geodesic compatibility and integrability of geodesic flows we give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a pseudo riemannian metric we prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition a lot of metrics studed in riemannian and kahlerian geometry satisfy such conditions we apply our results for obtaining an infinite family hierarchy of completely integrable flows on the complex projective plane cpn

the complex frobenius theorem for rough involutive structures we establish a version of the complex frobenius theorem in the context of a complex subbundle s of the complexified tangent bundle of a manifold having minimal regularity if the subbundle s defines the structure of a levi flat cr manifold it suffices that s be lipschitz for our results to apply a principal tool in the analysis is a precise version of the newlander nirenberg theorem with parameters for integrable almost complex structures with minimal regularity which builds on recent work of the authors