sasaki einstein manifolds and volume minimisation we study a variational problem whose critical point determines the reeb vector field for a sasaki einstein manifold this extends our previous work on sasakian geometry by lifting the condition that the manifolds are toric we show that the einstein hilbert action restricted to a space of sasakian metrics on a link l in a calabi yau cone x is the volume functional which in fact is a function on the space of reeb vector fields we relate this function both to the duistermaat heckman formula and also to a limit of a certain equivariant index on x that counts holomorphic functions both formulae may be evaluated by localisation this leads to a general formula for the volume function in terms of topological fixed point data as a result we prove that the volume of a sasaki einstein manifold relative to that of the round sphere is always an algebraic number in complex dimension n 3 these results provide via ads cft the geometric counterpart of a maximisation in four dimensional superconformal field theories we also show that our variational problem dynamically sets to zero the futaki invariant of the transverse space the latter being an obstruction to the existence of a kahler einstein metric

a homogeneous gibbons hawking ansatz and blaschke products abstract a homogeneous gibbons hawking ansatz is described leading to 4 dimensional hyperkahler metrics with homotheties in combination with blaschke products on the unit disc in the complex plane this ansatz allows one to construct infinite dimensional families of such hyperkahler metrics that are in a suitable sense complete our construction also gives rise to incomplete metrics on 3 dimensional contact manifolds that induce complete carnot caratheodory distances

eigenvalue boundary problems for the dirac operator on a compact riemannian spin manifold with mean convex boundary we analyse the ellipticity and the symmetry of four boundary conditions for the fundamental dirac operator including the global aps condition and a riemannian version of the local mit bag condition we show that friedrich s inequality for the eigenvalues of the dirac operator on closed spin manifolds holds for the corresponding four eigenvalue boundary problems more precisely we prove that for both the aps and the mit conditions the equality cannot be achieved and for the other two conditions the equality characterizes respectively half spheres and domains bounded by minimal hypersurfaces in manifolds carrying non trivial real killing spinors

a spinorial energy functional critical points and gradient flow let m be a compact spin manifold on the universal bundle of unit spinors we study a natural energy functional whose critical points if dim m ge 3 are precisely the pairs g varphi consisting of a ricci flat riemannian metric g together with a parallel g spinor varphi we investigate the basic properties of this functional and study its negative gradient flow the so called spinor flow in particular we prove short time existence and uniqueness for this flow

on the construction of some complete metrics with exceptional holonomy on construit explicitement trois metriques completes distinctes d holonomie egale a g 2 une metrique complete d holonomie egale a spin 7 et diverses metriques incompletes d holonomie exceptionnelle

prime fuzzy ideals in rings abstract in this paper give a characterization of prime fuzzy ideals we introduce weakly completely prime fuzzy ideals and establish a relation between completely prime fuzzy ideals and weakly completely prime fuzzy ideals

real ade equivariant co homotopy and super m branes a key open problem in m theory is the identification of the degrees of freedom that are expected to be hidden at ade singularities in spacetime comparison with the classification of d branes by k theory suggests that the answer must come from the right choice of generalized cohomology theory for m branes here we show that real equivariant cohomotopy on superspaces is a consistent such choice at least rationally after explaining this new approach we demonstrate how to use elmendorf s theorem in equivariant homotopy theory to reveal ade singularities as part of the data of equivariant s 4 valued super cocycles on 11d super spacetime we classify these super cocycles and find a detailed black brane scan that enhances the entries of the old brane scan to cascades of fundamental brane super cocycles on strata of intersecting black m brane species we find that on each singular stratum the black brane s instanton contribution namely its super nambu goto green schwarz action appears as the homotopy datum associated to the morphisms in the orbit category

extrinsic killing spinors under intrinsic and extrinsic curvature assumptions on a riemannian spin manifold and its boundary we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic killing spinors of non negative killing constant as a corollary we prove that a complete ricci flat spin manifold with mean convex boundary isometric to a round sphere is necessarily a flat disc

on the lattice of fuzzy ideals of a ring abstract first we prove that in the lattice of all fuzzy left right ideals with the same tip t using the technique of strong cut by this we obtain the following results 1 we point out that the lattice of all fuzzy ideals of a ring is not distributive thus the main theorem l f1 r is distributive in majumdar and sultana 1996 is erroneous 2 the lattice of all fuzzy left right ideals with same tip t of a ring r is modular this generalizes the main theorem in ajmal and thomas 1995

cones over pseudo riemannian manifolds and their holonomy by a classical theorem of gallot 1979 a riemannian cone over a complete riemannian manifold is either flat or has irreducible holonomy we consider metric cones with reducible holonomy over pseudo riemannian manifolds first we describe the local structure of the base of the cone when the holonomy of the cone is decomposable for instance we find that the holonomy algebra of the base is always the full pseudo orthogonal lie algebra one of the global results is that a cone over a compact and complete pseudo riemannian manifold is either flat or has indecomposable holonomy then we analyse the case when the cone has indecomposable but reducible holonomy which means that it admits a parallel isotropic distribution this analysis is carried out first in the case where the cone admits two complementary distributions and second for lorentzian cones we show that the first case occurs precisely when the local geometry of the base manifold is para sasakian and that of the cone

einstein matrices on s3 r3 and r4 bundles on construit a partir d un espace de base de kahler quaternionique 4n dimensionnel des metriques de cohomogeneite 1 en dimensions 4n 3 dont les surfaces de niveau sont l espace fibre s 2 des structures presque complexes sur la variete de base on obtient les conditions sur les fonctions metriques qui proviennent du fait qu on impose l equation d einstein et on obtient des solutions pour des espaces de dimension 4n 3 compactes et non compactes deux metriques 7 dimensionnelles ricci plates avec holonomie g 2 sont incluses dans les solutions non compactes

fuzzy nil radicals and fuzzy primary ideals abstract the purpose of this paper is to extend the concept of a fuzzy set by introducing the concept of a fuzzy nil radical of a fuzzy ideal in a ring using this concept a fuzzy primary ideal is defined some fundamental results concerning these notions are proved

solution of nonlinear volterra fredholm hammerstein integral equations via rationalized haar functions abstract rationalized haar functions are developed to approximate of the nonlinear volterra fredholm hammerstein integral equations the properties of rationalized haar functions are first presented and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

5d super yang mills on y p q sasaki einstein manifolds on any simply connected sasaki einstein five dimensional manifold one can construct a super yang mills theory which preserves at least two supersymmetries we study the special case of toric sasaki einstein manifolds known as y p q manifolds we use the localisation technique to compute the full perturbative part of the partition function the full equivariant result is expressed in terms of a certain special function which appears to be a curious generalisation of the triple sine function as an application of our general result we study the large n behaviour for the case of single hypermultiplet in adjoint representation and we derive the n 3 behaviour in this case

on fuzzy ideals of a ring 1 abstract the aim of this paper is to study fuzzy ideals of a ring to characterise regular rings and noetherian rings by fuzzy ideals and to determine all fuzzy prime ideals of the ring z of integers

half flat structures and special holonomy it was proven by hitchin that any solution of his evolution equations for a half flat su 3 structure on a compact six manifold m defines an extension of m to a seven manifold with holonomy in g2 we give a new proof which does not require the compactness of m more generally we prove that the evolution of any half flat g structure on a six manifold m defines an extension of m to a ricci flat seven manifold n for any real form g of sl 3 c if g is noncompact then the holonomy group of n is a subgroup of the noncompact form gof g c similar results are obtained for the extension of nearly half flat structures by nearly parallel g2 or g structures as well as for the extension of cocalibrated g2 and g structures by parallel spin 7 and spin0 3 4 structures respectively as an application we obtain that any six dimensional homogeneous manifold with an invariant half flat structure admits a canonical extension to a seven manifold with a parallel g2 or g structure for the group h3 h3 where h3 is the three dimensional heisenberg group we describe all left invariant half flat structures and develop a method to explicitly determine the resulting parallel g2 or g structure without integrating in particular we construct three eight parameter families of metrics with holonomy equal to g2 and g moreover we obtain a strong rigidity result for the metrics induced by a half flat structure on h3 h3 satisfying z z 0 where z denotes the centre finally we describe the special geometry of the space of stable three forms satisfying a reality condition considering all possible reality conditions we find four different special kahler manifolds and one special para kahler manifold

pseudoholomorphic curves in symplectizations with applications to the weinstein conjecture in dimension three l in t roduct ion 515 1 1 the weinstein conjecture 515 1 2 the main results 517 1 3 sketch of the proof 522 2 local f redho lm theory 525 2 1 similar i ty principle and consequences 525 2 2 n m as a lmost complex manifold and local fillings 529 2 3 local propert ies of ho lomorphic curves 531 3 bubbl ing off analysis and global fredholm theory 533 3 l grad ien t bounds imply ca bounds 533 3 2 bubbing off analysis of ho lomorphic planes 534 3 3 grad ien t bounds near the boundary 537 4 proof of the main results 544 4 1 set up for a global fredholm theory 544 4 2 bishop s theorem 545 4 3 proof of theorem 5 546 4 4 proof of theorems 8 and 11 547 4 5 uniqueness results for foliations 547 5 appendix 550 5 1 per turbing surfaces 550 5 2 ki l l ing crit ical points of hypersurfaces 552 5 3 adding a thin tube to a sphere 560

imaginary killing spinors in lorentzian geometry we study the geometric structure of lorentzian spin manifolds which admit imaginary killing spinors the discussion is based on the cone construction and a normal form classification of skew adjoint operators in signature 2 n 2 derived geometries include brinkmann spaces lorentzian einstein sasaki spaces and certain warped product structures exceptional cases with decomposable holonomy of the cone are possible

the lattice of fuzzy ideals of a ring the set of all fuzzy ideals of a ring r is partially ordered with respect to the natural ordering of fuzzy sets in r it has been shown that this is a lattice but not a sublattice of the lattice of all fuzzy sets in r this lattice of all fuzzy ideals of r is complete distributive and bounded and has neither atoms nor dual atoms

fuzzy ideals and fuzzy prime ideals of a ring abstract besides obtaining certain ring theoretic analogues some peculiarities of the fuzzy setting are also discussed in particular a study of i the conditions under which a given fuzzy ideal can or cannot be expressed as a union of two proper fuzzy ideals ii the algebraic nature of images and preimages of fuzzy prime ideals under homomorphisms and iii the ring of all the fuzzy cosets of a fuzzy prime fuzzy maximal ideal is carried out

new g _2 holonomy cones and exotic nearly kahler structures on s 6 and s 3 times s 3 there is a rich theory of so called strict nearly kaehler manifolds almost hermitian manifolds generalising the famous almost complex structure on the 6 sphere induced by octonionic multiplication nearly kaehler 6 manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional lie group g 2 the metric cone over a riemannian 6 manifold m has holonomy contained in g 2 if and only if m is a nearly kaehler 6 manifold a central problem in the field has been the absence of any complete inhomogeneous examples we prove the existence of the first complete inhomogeneous nearly kaehler 6 manifolds by proving the existence of at least one cohomogeneity one nearly kaehler structure on the 6 sphere and on the product of a pair of 3 spheres we conjecture that these are the only simply connected inhomogeneous cohomogeneity one nearly kaehler structures in six dimensions

embedding lattices of fuzzy subalgebras into lattices of crisp subalgebras abstract given a universal algebra a we define an algebra a where a and a are of the same type and generate the same variety of algebra next we define a map e lf a lc a where lf a is the lattice of fuzzy subalgebras of a and lc a is the lattice of crisp subalgebras of a for many algebras a the map e is shown to be a lattice embedding as a corollary of this fact we derive the results that the lattice of fuzzy normal subgroups of a group and the lattice of fuzzy ideals of a ring are both modular lattices

7 dimensional compact riemannian manifolds with killing spinors using a link between einstein sasakian structures and killing spinors we prove a general construction principle of odd dimensional riemannian manifolds with real killing spinors in dimensionn 7 we classify all compact riemannian manifolds with two or three killing spinors finally we classify nonflat 7 dimensional riemannian manifolds with parallel spinor fields

real killing spinors and holonomy we give a description of all complete simply connected riemannian manifolds carrying real killing spinors furthermore we present a construction method for manifolds with the exceptional holonomy groupsg2 and spin 7

kahlerian twistor spinors on a kahler spin manifold kahlerian twistor spinors are a natural analogue of twistor spinors on riemannian spin manifolds they are defined as sections in the kernel of a first order differential operator adapted to the kahler structure called kahlerian twistor penrose operator we study kahlerian twistor spinors and give a complete description of compact kahler manifolds of constant scalar curvature admitting such spinors as in the riemannian case the existence of kahlerian twistor spinors is related to the lower bound of the spectrum of the dirac operator

fuzzy semiprimary ideals of rings abstract the concept of fuzzy ideals is extended by introducing fuzzy semiprimary ideals in rings this class of fuzzy ideals generalizes the class of fuzzy semiprimary ideals it is shown that there do exist fuzzy ideals which are fuzzy semiprimary but not fuzzy primary a study of i the level ideals of a fuzzy semiprimary ideal ii the characteristic function of a semiprimary ideal iii the zero divisors in the ring of all fuzzy cosets of a fuzzy semiprimary ideal and iv the algebraic nature of direct and inverse images of fuzzy semiprimary ideals under homomorphisms is carried out

fuzzy irreducible ideals in rings abstract in this paper the concept of a fuzzy set is extended by introducing the concept of a fuzzy irreducible ideal in a ring the characteristic function of an irreducible ideal and the level ideals of a fuzzy irreducible ideal are studied the algebraic nature of homomorphic images and preimages of fuzzy irreducible ideals is also examined some other fundamental results pertaining to this concept are discussed

fuzzy prime ideals of a ring abstract this paper characterizes all fuzzy prime ideals p of an arbitrary ring r we show that a nonconstant fuzzy ideal p of r is prime if and only if p0 x r p x p 0 is a prime ideal of r p is two valued and p 0 1 examples are given showing that p0 is a prime ideal is not sufficient for p to be a fuzzy prime ideal and that p0 may be a prime ideal of r with the image of p being of arbitrary finite cardinality or being countably infinite

representation of exact solution for the nonlinear volterra fredholm integral equations abstract in this paper the representation of the exact solution for the nonlinear volterra fredholm integral equations will be obtained in the reproducing kernel space the exact solution is given by the form of series its approximate solution is obtained by truncating the series and a new numerical approximate method is obtained the error of the approximate solution is monotone deceasing in the sense of w 2 1 a b the intrinsic merit of the method given in the paper lies in its speedy convergence in addition it must be pointed out that additional conditions as linearization or unjustified assumptions are not required for nonlinear cases

fuzzy prime ideals of rings abstract the notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers a procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established all fuzzy prime maximal ideals of a given ring are determined by establishing a one to one correspondence between fuzzy prime maximal ideals and the pairs p where p is a prime maximal ideal of the ring and is a prime element dual atom in the complete lattice

numerical solution of nonlinear volterra fredholm hammerstein integral equations via collocation method based on radial basis functions abstract a numerical technique based on the spectral method is presented for the solution of nonlinear volterra fredholm hammerstein integral equations this method is a combination of collocation method and radial basis functions rbfs with the differentiation process drbf using zeros of the shifted legendre polynomial as the collocation points different applications of rbfs are used for this purpose the integral involved in the formulation of the problems are approximated based on legendre gauss lobatto integration rule the results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme

fuzzy invariant subgroups and fuzzy ideals abstract the purpose of this paper is to introduce some basic concepts of fuzzy algebra as fuzzy invariant subgroups fuzzy ideals and to prove some fundamental properties in particular it will give a characteristic of a field by a fuzzy ideal

the twistor equation in lorentzian spin geometry in this paper we discuss the twistor equation in lorentzian spin geometry in particular we explain the local conformal structure of lorentzian manifolds which admit twistor spinors inducing lightlike dirac currents furthermore we derive all local geometries with singularity free twistor spinors that occur up to dimension 7

embedding into manifolds with torsion we introduce a class of special geometries associated to the choice of a differential graded algebra contained in lambda mathbb r n we generalize some known embedding results that effectively characterize the real analytic riemannian manifolds that can be realized as submanifolds of a riemannian manifold with special holonomy to this more general context in particular we consider the case of hypersurfaces inside nearly kahler and einstein sasaki manifolds proving that the corresponding evolution equations always admit a solution in the real analytic case

he s variational iteration method for solving nonlinear mixed volterra fredholm integral equations in this paper he s variational iteration method is used to give the approximate solution of nonlinear mixed volterra fredholm integral equations the method constructs a convergent sequence of functions which approximates the exact solution with few iterations numerical results and a comparison with the exact solution are given which reveal its efficiency

a reliable treatment for mixed volterra fredholm integral equations the main goal of this paper is to demonstrate the use of the modified decomposition method for mixed nonlinear volterra fredholm integral equations the modified method combined with the noise terms phenomena may provide the exact solution by using two iterations only two numerical illustrations are given to show the pertinent features of the technique

parallel spinors and holonomy groups in this paper we complete the classification of spin manifolds admitting parallel spinors in terms of the riemannian holonomy groups more precisely we show that on a given n dimensional riemannian manifold spin structures with parallel spinors are in one to one correspondence with lifts to spinn of the riemannian holonomy group with fixed points on the spin representation space in particular we obtain the first examples of compact manifolds with two different spin structures carrying parallel spinors

metrics with exceptional holonomy it is proved that there exist metrics with holonomy g2 and spin 7 thus settling the remaining cases in berger s list of possible holonomy groups we first reformulate the holonomy h condition as a set of differential equations for an associated h structure on a given manifold we collect the needed algebraic facts about g2 and spin 7 we then apply the machinery of over determined partial differential equations in the form of the cartan kahler theorem to prove the existence of solutions whose holonomy is g2 or spin 7 we also provide explicit examples and some information about the generality of the space of

integrability conditions for killing spinors the conditions for the existence of solutions ofd c are discussed in general it is not sufficient to consider only the first integrability condition d dv 2c2 v in particular the second integrability condition is needed to explain why in certain cases only for one choice of sign does a solution exist the killing spinor tensors as defined by walker and penrose are shown to be the spinorial equivalent of conformal killing tensors their relationship to the killing spinors and spinor vectors used in supergravity is given

taylor polynomial solutions of nonlinear volterra fredholm integral equations the kanwal and liu method for the solution of fredholm integral equations is applied to certain nonlinear volterra fredholm integral equations in addition examples that illustrate the pertinent features of the method are presented and the results of study are discussed

a characterization of l fuzzy prime ideals abstract in this paper the concepts of a l fuzzy prime and a l fuzzy completely prime ideal are given and some fundamental lemmas are proved in particular bijections among some of the l fuzzy prime ideals of two homomorphic rings are constructed also a characterization of a l fuzzy prime ideal of a ring is given

certain fuzzy ideals of rings redefined abstract an attempt is made to redefine fuzzy semiprimary fuzzy primary fuzzy prime and fuzzy maximal ideals of rings these fuzzy ideals are characterized by their level ideals fuzzy analogues of various results concerning semiprimary primary prime and maximal ideals are obtained the concept of a fuzzy nil radical of a fuzzy ideal is also formulated and discussed

prime l fuzzy ideals and primary l fuzzy ideals abstract in this paper the concepts of a primary l fuzzy ideal and a prime l fuzzy ideal and p primary are introduced and some fundamental propositions proved in particular a structural theorem for a primary l fuzzy ideal belonging to a prime l fuzzy ideal is given

the lattice of l ideals of a ring is modular in this paper we extend the notion of a tip extended pair of fuzzy subgroups to l ideals of a ring we prove that the sum of two tip extended l ideals of an arbitrary pair of l ideals of a ring is the least l ideal containing the union of the given l ideals using this construction of join of l ideals we prove that the lattice of all l ideals of a given ring is modular

the first eigenvalue of the dirac operator on quaternionic kahler manifolds in 15 we proved a lower bound for the spectrum of the dirac operator on quaternionic kahler manifolds in the present article we show that the only manifolds in the limit case i e the only manifolds where the lower bound is attained as an eigenvalue are the quaternionic projective spaces we use the equivalent formulation in terms of the quaternionic killing equation introduced in 16 and show that a nontrivial solution defines a parallel spinor on the associated hyperkahler manifold

einstein metrics on s 3 r 3 and r 4 bundles starting from a 4n dimensional quaternionic kahler base space we construct metrics of cohomogeneity one in 4n 3 dimensions whose level surfaces are thes 2 bundle space of almost complex structures on the base manifold we derive the conditions on the metric functions that follow from imposing the einstein equation and obtain solutions both for compact and non compact 4n 3 dimensional spaces included in the non compact solutions are two ricci flat 7 dimensional metrics withg 2 holonomy we also discuss two other ricci flat solutions one on ther 4 bundle overs 3 and the other on anr 4 bundle overs 4 these haveg 2 and spin 7 holonomy respectively

einstein manifolds of dimension five with small first eigenvalue of the dirac operator soit m n une variete de spin compacte d einstein a courbure scalaire positive r 0 et on note par d t s t s l operateur de dirac agissant sur des sections du fibre spinoriel on etudie le probleme de la classification des varietes pour le cas a 5 dimensions

generalized cylinders in semi riemannian and spin geometry we use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory it has the advantage of being very simple and the result directly extends to semi riemannian manifolds and to embeddings into spaces of constant curvature we also give a new way to identify spinors for different metrics and to derive the variation formula for the dirac operator moreover we show that generalized killing spinors for codazzi tensors are restrictions of parallel spinors finally we study the space of lorentzian metrics and give a criterion when two lorentzian metrics on a manifold can be joined in a natural manner by a 1 parameter family of such metrics

la premiere valeur propre de l operateur de dirac sur les varietes kahleriennes compactes k d kirchberg ki1 gave a lower bound for the first eigenvalue of the dirac operator on a spin compact kahler manifoldm of odd complex dimension with positive scalar curvature we prove that manifolds of real dimension 8l 6 satisfying the limiting case are twistor space cf sa of quaternionic kahler manifold with positive scalar curvature and that the only manifold of real dimension 8l 2 satisfying the limiting case is the complex projective spacecp4l 1

reeb dynamics inspired by katok s example in finsler geometry inspired by katok s examples of finsler metrics with a small number of closed geodesics we present two results on reeb flows with finitely many periodic orbits the first result is concerned with a contact geometric description of magnetic flows on the 2 sphere found recently by benedetti we give a simple interpretation of that work in terms of a quaternionic symmetry in the second part we use hamiltonian circle actions on symplectic manifolds to produce compact connected contact manifolds in dimension at least five with arbitrarily large numbers of periodic reeb orbits this contrasts sharply with recent work by cristofaro gardiner hutchings and pomerleano on reeb flows in dimension three with the help of hamiltonian plugs and a surgery construction due to laudenbach we reprove a result of cieliebak one can produce hamiltonian flows in dimension at least five with any number of periodic orbits in dimension three with any number greater than one

contact spheres and hyperkahler geometry a taut contact sphere on a 3 manifold is a linear 2 sphere of contact forms all defining the same volume form in the present paper we completely determine the moduli of taut contact spheres on compact left quotients of su 2 the only closed manifolds admitting such structures we also show that the moduli space of taut contact spheres embeds into the moduli space of taut contact circles

an efficient algorithm for solving nonlinear volterra fredholm integral equations in this work we develop a new effective method for solving nonlinear volterra fredholm integral equation the existence of any e approximate solution is proved at the same time an effective method for obtaining the e approximate solution is established the final numerical examples illustrate that our approach is valid not only for weakly nonlinear problems but also for strongly nonlinear problems

numerical solutions of the nonlinear volterra fredholm integral equations by using homotopy perturbation method abstract a numerical method for solving the nonlinear volterra fredholm integral equations is presented the method is based upon homotopy perturbation theory the results reveal that the homotopy perturbation method is very effective and simple and gives the exact solution

nambu jacobi and generalized jacobi manifolds the geometry of nambu jacobi and generalized jacobi manifolds is studied a large collection of examples is given the characteristic distribution generated by the hamiltonian vector fields on a nambu jacobi manifold is proved to be completely integrable and the induced geometrical structure of the leaves of the corresponding generalized foliation is ellucidated

cohomogeneity one einstein sasaki 5 manifolds we consider hypersurfaces in einstein sasaki 5 manifolds which are tangent to the characteristic vector field we introduce evolution equations that can be used to reconstruct the 5 dimensional metric from such a hypersurface analogous to the nearly hypo and half flat evolution equations in higher dimensions we use these equations to classify einstein sasaki 5 manifolds of cohomogeneity one

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

generic metrics and connections on spin and spin c manifolds we study the dependence of the dimension h0 g a of the kernel of the atyiah singer dirac operator cal d g a on a spinc manifold m on the metric g and the connection a the main result is that in the case of spin structures the value of h0 g for the generic metric is given by the absolute value of the index provided dim m in 3 4 in dimension 2 the mod 2 index theorems have to be taken into a account and we obtain an extension of a classical result in the theory of riemann surfaces in the spinc case we also discuss upper bounds on h0 g a for generic metrics and we obtain a complete result in dimension 2 the much simpler dependence on the connection a and applications to seiberg witten theory are also discussed