generic hkt geometries in the harmonic superspace approach we explain how a generic hyper kahler with torsion hkt geometry can be derived using the language of n 4 supersymmetric quantum mechanics to this end one should consider a lagrangian involving several 4 4 0 multiplets defined in harmonic superspace and subject to nontrivial harmonic constraints conjecturally this general construction presented in the paper of f delduc and e ivanov nucl phys b 855 815 2012 gives a complete classification of all hkt geometries each such geometry is generated by two different functions potentials of a special type that depend on harmonic superfields and on harmonics given these two potentials one can derive the vielbeins metric connections and curvatures but this is not so simple one should solve rather complicated differential equations we illustrate the general construction by giving a detailed derivation of the metric for hyper kahler taub nut manifolds in the generic case we arrive at an hkt geometry in this paper we give a simple proof

connections with exotic holonomy connections with exotic holonomy lorenz johannes schwachhofer wolfgang ziller supervisor in 1955 berger ber partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold however it was shown by bryant br2 that berger s list is incomplete connections whose holonomy is not contained on berger s list are called exotic we investigate examples of a certain 4 dimensional exotic holonomy representation of sl 2 r we show that connections with this holonomy are never complete give explicit descriptions of these connections on an open dense set and compute their group of symmetry finally we give strong restrictions for their existence on compact manifolds

compact riemannian 7 manifolds with holonomy g sb 2 ii this is the second of two papers about metrics of holonomy g2 on compact 7manifolds in our first paper 15 we established the existence of a family of metrics of holonomy g2 on a single compact simply connected 7 manifold m using three general results theorems a b and c our purpose in this paper is to explore the theory of compact riemannian 7 manifolds with holonomy g2 in greater detail by relying on theorems a c we will be able to avoid the emphasis on analysis that characterized 15 so that this sequel will have a more topological flavour the paper has four chapters the first chapter consists of introductory material section 1 1 gives some elementary geometric and topological material on compact 7 manifolds with torsion free g2structures then 1 2 describes the holonomy groups su 2 and su 3 and 1 3 explains the concept of asymptotically locally euclidean riemannian manifolds shortened to ale spaces with special holonomy recall that in 15 a compact 7 manifold m was defined by desingularizing a quotient t of the 7 torus by a finite group of isometries z2 the subject of chapters 2 and 3 is a generalization of this idea chapter 2 defines a general construction for compact 7 manifolds with torsion free g2structures which works by desingularizing quotients t for finite groups the ale spaces with holonomy su 2 and su 3 discussed in 1 3 are an essential ingredient in performing this desingularization the central result of chapter 2 is theorem 2 2 3 which states that given a suitable finite group and certain other data one may construct a compact 7manifold m from t that admits torsion free g2structures this result is proved using theorems a c of 15 chapter 3 is devoted entirely to examples of this construction we give many examples of compact 7 manifolds with holonomy g2 and determine their basic topological invariants the betti numbers and fundamental group finally in chapter 4 we discuss some areas of interest and give a number of open problems this paper is not written to be read independently of 15 the language and results of 15 will be used freely in particular the introductory material in 15 1 1 for reference we reproduce here the model 3and 4forms defining the flat g2structure on r as given in 15 1 1

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

properties of manifolds with skew symmetric torsion and special holonomy abstract in this paper we provide examples of hypercomplex manifolds which do not carry hkt structures thus answering a question in grantcharov and poon comm math phys 213 2000 19 we also prove that the existence of an hkt structure is not stable under small deformations similarly we provide examples of compact complex manifolds with vanishing first chern class which do not admit a hermitian structure whose bismut connection has restricted holonomy in su n thus providing a counter example to the conjecture in gutowski et al deformations of generalized calibrations and compact non kahler manifolds with vanishing first chern class math dg 0205012 asian j math to appear again we prove that such a property is not stable under small deformations

deformations in g2 manifolds here we study the deformations of associative submanifolds inside a g2 manifold m7 with a calibration 3 form f a choice of 2 plane field on m which always exists splits the tangent bundle of m as a direct sum of a 3 dimensional associate bundle and a complex 4 plane bundle tm ev and this helps us to relate the deformations to seiberg witten type equations here all the surveyed results as well as the new ones about g2 manifolds are proved by using only the cross product operation equivalently f we feel that mixing various different local identifications of the rich g2 geometry e g cross product representation theory and the algebra of octonions makes the study of g2 manifolds look harder then it is e g the proof of mclean s theorem r c mclean deformations of calibrated submanifolds comm anal geom 6 1998 705 747 we believe the approach here makes things easier and keeps the presentation elementary

e_7 wirtinger inequalities cayley 4 form and homotopy we study optimal curvature free inequalities of the type discovered by c loewner and m gromov using a generalisation of the wirtinger inequality for the comass using a model for the classifying space bs built inductively out of bs we prove that the symmetric metrics of certain two point homogeneous manifolds turn out not to be the systolically optimal metrics on those manifolds we point out the unexpected role played by the exceptional lie algebra e7 in systolic geometry via the calculation of wirtinger constants using a technique of pullback with controlled systolic ratio we calculate the optimal systolic ratio of the quaternionic projective plane modulo the existence of a joyce manifold with spin 7 holonomy and unit middle dimensional betti number

improved mixing time bounds for the thorp shuffle and l reversal chain we prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards then we use it to obtain improved bounds for two previously studied models e thorp introduced the following card shuffling model in 1973 suppose the number of cards n is even cut the deck into two equal piles drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip then drop from the other pile continue this way until both piles are empty we obtain a mixing time bound of o log 4 n previously the best known bound was o log 29 n and previous proofs were only valid for n a power of 2 we also analyze the following model called the l reversal chain introduced by durrett there are n cards arrayed in a circle each step an interval of cards of length at most l is chosen uniformly at random and its order is reversed durrett has conjectured that the mixing time is o max n n 3 l 3 log n we obtain a bound that is within a factor o log 2 n of this the first bound within a poly log factor of the conjecture

on the complex cayley grassmannian abstract we define a torus action on the complex cayley grassmannian x using this action we prove that x is a singular variety we also show that the singular locus is smooth and has the same cohomology ring as that of c p 5 furthermore we identify the singular locus with a quotient of g 2 c by a parabolic subgroup

compact 8 manifolds with holonomy spin 7 in berger s classification 4 of the possible holonomy groups of a nonsymmetric irreducible riemannian manifold there are two special cases the exceptional holonomy groups g2 in 7 dimensions and spin 7 in 8 dimensions bryant 6 proved that such metrics exist locally using the theory of exterior differential systems and gave some explicit examples later bryant and salamon 7 constructed explicit complete metrics with holonomy g2 and spin 7 in two previous papers 12 13 the author constructed many examples of compact riemannian 7 manifolds with holonomy g2 this paper will construct examples of compact riemannian 8 manifolds with holonomy spin 7 using similar methods we believe that these are the first examples known since metrics with holonomy spin 7 are ricci flat these are also new examples of compact ricci flat riemannian 8 manifolds let m be an 8 manifold a spin 7 structure on m can be encoded in a 4 form on m a special 4 form satisfying the condition that the stabilizer of at each point should be isomorphic to spin 7 by an abuse of notation we usually identify the spin 7 structure with its associated 4 form since spin 7 so 8 the spin 7 structure also induces a riemannian metric g and an orientation on m it turns out that the holonomy group hol g of g is a subgroup of spin 7 with spin 7 structure if and only if d 0 the quantity d is called the torsion of the spin 7 structure and is called torsion free if d 0 very briefly the plan of the paper splits into four steps as follows firstly a compact 8 manifold m is given secondly a spin 7 structure on m is found with small torsion i e d is small thirdly is deformed to a nearby spin 7 structure with d 0 thus is torsion free and if g is the associated metric then hol g spin 7 fourthly it is shown that hol g is spin 7 and not some proper subgroup the structure of the paper is designed around this division of the construction into four steps there are six chapters this first chapter is of introductory material the second chapter has the full statements of the main results in

g2 structures and octonion bundles abstract we use a g 2 structure on a 7 dimensional riemannian manifold with a fixed metric to define an octonion bundle with a fiberwise non associative product we then define a metric compatible octonionic covariant derivative on this bundle that is compatible with the octonion product the torsion of the g 2 structure is then shown to be an octonionic connection for this covariant derivative with curvature given by the component of the riemann curvature that lies in the 7 dimensional representation of g 2 we also interpret the choice of a particular g 2 structure within the same metric class as a choice of gauge and show that under a change of this gauge the torsion does transform as an octonion valued connection 1 form finally we also show an explicit relationship between the octonion bundle and the spinor bundle define an octonionic dirac operator and explore an energy functional for octonion sections we then prove that critical points correspond to divergence free torsion which is shown to be an octonionic analog of the coulomb gauge

a heat flow for special metrics on the space of positive 3 forms on a seven manifold we study a natural functional whose critical points induce metrics with holonomy contained in g2 we prove short time existence and uniqueness for its negative gradient flow furthermore we show that the flow exists for all times and converges modulo dieomorphisms to some critical point for any initial condition suciently c 1 close to a critical point

disk envelopes of functions ii given a suslin function on a domainyin a banach spacexone can define two envelopes of related to the pluripotential theory the first one denoted byp is the supremum of all plurisubharmonic functions less than the value of the other one denoted byd at some pointzis equal to the infimum of integrals over the boundaries of all closed analytic disks centered atz in this paper we study the relationship between these envelopes we show thatd p for a large class of functions on domains in banach spaces wherep is the upper regularization ofp ifx cn then we prove thatd is always equal top

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

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

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

duality and yang mills fields on quaternionic kahler manifolds the concept of a self dual connection on a four dimensional riemannian manifold is generalized to the 4n dimensional case of any quaternionic kahler manifold the generalized self dual connections are minima of a modified yang mills functional it is shown that our definitions give a correct framework for a mapping theory of quaternionic kahler manifolds the mapping theory is closely related to the construction of yang mills fields on such manifolds some monopole like equations are discussed

on the geometry of closed g 2 structures we give an answer to a question posed in physics by cvetic et al 9 and recently in mathematics by bryant 3 namely we show that a compact 7 dimensional manifold equipped with a g2 structure with closed fundamental form is einstein if and only if the riemannian holonomy of the induced metric is contained in g2 this could be considered to be a g2 analogue of the goldberg conjecture in almost kahler geometry and was indicated by cvetic et al in 9 the result was generalized by bryant to closed g2 structures with too tightly pinched ricci tensor we extend it in another direction proving that a compact g2 manifold with closed fundamental form and divergence free weyl tensor is a g2 manifold with parallel fundamental form we introduce a second symmetric ricci type tensor and show that einstein conditions applied to the two ricci tensors on a closed g2 structure again imply that the induced metric has holonomy group contained in g2

the holonomy lie algebras of neutral metrics in dimension four the lie algebra isomorphism between su 1 1 su 1 1 and o 2 2 is used to obtain a list of subalgebras of the latter the resulting list of 32 subalgebras is then examined on a case by case basis to see if each can be the lie algebra of the holonomy group of a neutral metric in four dimensions the conclusions taken in conjunction with previously known results furnish a classification of such lie subalgebras of o 2 2 with only one case remaining unresolved

mixing of the symmetric exclusion processes in terms of the corresponding single particle random walk we prove an upper bound for the e mixing time of the symmetric exclusion process on any graph g with any feasible number of particles our estimate is proportional to trw g ln v e where v is the number of vertices in g and trw g is the 1 4 mixing time of the corresponding single particle random walk this bound implies new results for symmetric exclusion on expanders percolation clusters the giant component of the erdos renyi random graph and poisson point processes in rd our technical tools include a variant of morris s chameleon process

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

singular lines of trilinear forms abstract we prove that an alternating e form on a vector space over a quasi algebraically closed field always has a singular e 1 dimensional subspace provided that the dimension of the space is strictly greater than e here an e 1 dimensional subspace is called singular if pairing it with the e form yields zero by the theorem of chevalley and warning our result applies in particular to finite base fields our proof is most interesting in the case where e 3 and the space has odd dimension n then it involves a beautiful equivariant map from alternating trilinear forms to polynomials of degree n 1 2 1 we also give a sharp upper bound on the dimension of subspaces all of whose two dimensional subspaces are singular for a non degenerate trilinear form in certain binomial dimensions the trilinear forms attaining this upper bound turn out to form a single orbit under the general linear group and we classify their singular lines

dirichlet problems for plurisubharmonic functions on compact sets in this paper we solve the dirichlet problems for different classes of plurisubharmonic functions on compact sets in mathbb c n including continuous pluriharmonic and maximal functions

on the existence of infinite series of exotic holonomies in 1955 berger ber gave a list of irreducible reductive representations which can occur as the holonomy of a torsion free affine connection while this list was stated to be complete in the case of metric connections the situation in the general case remained unclear the representations which are missing from this list are called exotic in this paper we use twistor techniques to detect an infinite series of candidates for exotic holonomies we then develop a general method for constructing torsion free affine connections with prescribed holonomy which is based onequivariant deformations of a certain class of linear poisson structures when applied to the new series this method yields an exhaustive description of all torsion free connections with these holonomies and hence not only proves the existence of such connections but also allows us to deduce some striking facts about their local and global behaviour

on the stability of riemannian manifold with parallel spinors inspired by the recent work hhm03 we prove two stability results for compact riemannian manifolds with nonzero parallel spinors our first result says that ricci flat metrics which also admit nonzero parallel spinors are stable in the direction of changes in conformal structures as the critical points of the total scalar curvature functional our second result which is a local version of the first one shows that any metric of positive scalar curvature cannot lie too close to a metric with nonzero parallel spinor we also prove a rigidity result for special holonomy metrics in the case of su m holonomy the rigidity result implies that scalar flat deformations of calabi yau metric must be calabi yau finally we explore the connection with a positive mass theorem of d03 which presents another approach to proving these stability and rigidity results

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

chiral de rham complex on riemannian manifolds and special holonomy interpreting the chiral de rham complex cdr as a formal hamiltonian quantization of the supersymmetric non linear sigma model we suggest a setup for the study of cdr on manifolds with special holonomy we show how to systematically construct global sections of cdr from differential forms and investigate the algebra of the sections corresponding to the covariantly constant forms associated with the special holonomy as a concrete example we construct two commuting copies of the odake algebra an extension of the n 2 superconformal algebra on the space of global sections of cdr of a calabi yau threefold and conjecture similar results for g2 manifolds we also discuss quasi classical limits of these algebras

kahler reduction of metrics with holonomy g2 a torsion free g 2 structure admitting an infinitesimal isometry such that the quotient is a kahler manifold is shown to give rise to a 4 manifold equipped with a complex symplectic structure and a 1 parameter family of functions and 2 forms linked by second order equations reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to g 2

holonomy groups and w symmetries irreducible sigma models i e those for which the partition function does not factorise are defined on riemannian spaces with irreducible holonomy groups these special geometries are characterised by the existence of covariantly constant forms which in turn give rise to symmetries of the supersymmetric sigma model actions the poisson bracket algebra of the corresponding currents is aw algebra extended supersymmetries arise as special cases

exceptional holonomy and einstein metrics constructed from aloff wallach spaces we investigate cohomogeneity one metrics whose principal orbit is an aloff wallach space su 3 u 1 in particular we are in terested in metrics whose holonomy is contained in spin 7 complete metrics of this kind which are not product metrics have exactly one singular orbit we prove classification results for metrics on tubular neighborhoods of various singular orbits since the equation for the ho lonomy reduction has only few explicit solutions we make use of power series techniques in order to prove the convergence and the smoothness near the singular orbit we apply methods developed by eschenburg and wang as a by product of these methods we find many new examples of einstein metrics of cohomogeneity one

su 3 instantons and g 2 spin 7 heterotic string solitons necessary and sufficient conditions to the existence of a hermitian connection with totally skew symmetric torsion and holonomy contained in su 3 are given a formula for the riemannian scalar curvature is obtained non compact solution to the supergravity type i equations of motion with non zero flux and non constant dilaton is found in dimension 6 non conformally flat non compact solutions to the supergravity type i equations of motion with non zero flux and non constant dilaton are found in dimensions 7 and 8 a riemannian metric with holonomy contained in g2 arises from our considerations and hitchin s flow equations which seems to be new compact examples of su 3 g2 and spin 7 instanton satisfying the anomaly cancellation conditions are presented

multi moment maps we introduce a notion of moment map adapted to actions of lie groups that preserve a closed three form we show existence of our multi moment maps in many circumstances including mild topological assumptions on the underlying manifold such maps are also shown to exist for all groups whose second and third lie algebra betti numbers are zero we show that these form a special class of solvable lie groups and provide a structural characterisation we provide many examples of multi moment maps for different geometries and use them to describe manifolds with holonomy contained in g2 preserved by a two torus symmetry in terms of tri symplectic geometry of four manifolds

approximation by harmonic functions for a compact set x rn we construct a restoring covering for the space h x of real valued functions on x which can be uniformly approximated by harmonic functions functions from h x restricted to an element y of this covering possess some analytic properties in particular every nonnegative function f h y equal to 0 on an open non void set is equal to 0 on y moreover when n 2 the algebra h y of complexvalued functions on y which can be uniformly approximated by holomorphic functions is analytic these theorems allow us to prove that if a compact set x c has a nontrivial jensen measure then x contains a nontrivial compact set y with analytic algebra h y

approximation of jensen measures by image measures under holomorphic functions and applications we show that jensen measures defined on c n or more generally on a complex banach space x can be approximated by the image of lebesgue measure on the torus under x valued polynomials defined on c we give similar characterizations for jensen measures in terms of analytic martingales and hardy martingales the results are applied to approximate plurisubharmonic martingales by hardy martingales which enables us to give a characterization of the analytic radon nikodym property of banach spaces in terms of convergence of plurisubharmonic martingales thus solving a problem of g a edgar

geometry of hyper kahler connections with torsion the internal space of a n 4 supersymmetric model with wess zumino term has a connection with totally skew symmetric torsion and holonomy in sp n we study the mathematical background of this type of connection in particular we relate it to classical hermitian geometry construct homogeneous as well as inhomogeneous examples characterize it in terms of holomorphic data develop its potential theory and reduction theory

mirror duality in a joyce manifold abstract previously the two of the authors defined a notion of dual calabi yau manifolds in a g 2 manifold and described a process to obtain them here we apply this process to a compact g 2 manifold constructed by joyce and as a result we obtain a pair of borcea voisin calabi yau manifolds which are known to be mirror duals of each other

a note on exceptional groups and reidemeister torsion let be a closed orientable surface of genus at least 2 and g be one of the exceptional groups g2 f4 and e6 the present article considers the set rep g of g valued representations from the fundamental group 1 of the surface to the exceptional group g it proves that for such representations the notion of reidemeister torsion is well defined it also establishes a formula for computing reidemeister torsion of such representations in terms of the well known symplectic structure on rep g namely the atiyah bott goldman symplectic form for the lie group g moreover it applies to g valued hitchin representations

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

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

lie orthogonal operators abstract lie orthogonal operators on a finite dimensional lie algebra are studied the components of the ascending central series and the radical are invariant with respect to any lie orthogonal operator only solvable lie algebras with solvability degree not greater than two admit lie orthogonal without eigenvalues 1 and 1 lie orthogonal operators on a simple lie algebra are trivial this gives the description of lie orthogonal operators for semi simple and reductive algebras lie orthogonal operators of some classes of lie algebras are directly computed thus the equivalence classes of lie orthogonal operators on a heisenberg algebra form the symplectic group of an appropriate dimension

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

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

minimal submanifolds of ⁿ arising from degenerate 3 orbits on the grassmannian we give new examples of minimal submanifolds of e 2n characterised by having their gauss map s image lie in degenerate s0 3 orbits of gp2n 1 the grassmannian of p planes in e where the action on 2n 1 n gp 2n 1 is induced from the irreducible s0 3 action on r2n these submanifolds are all given explicitly in terms of holomorphic data and are linearly 2n l1 full in e

new hkt manifolds arising from quaternionic representations we give a procedure for constructing an 8n dimensional hkt lie algebra starting from a 4n dimensional one by using a quaternionic representation of the latter the strong respectively weak hyper kahler balanced condition is preserved by our construction as an application of our results we obtain a new compact hkt manifold with holonomy in sl n mathbb h which is not a nilmanifold we find in addition new compact strong hkt manifolds we also show that every kahler lie algebra equipped with a flat torsion free complex connection gives rise to an hkt lie algebra we apply this method to two distinguished 4 dimensional kahler lie algebras thereby obtaining two conformally balanced hkt metrics in dimension 8 both techniques prove to be an effective tool for giving the explicit expression of the corresponding hkt metrics

harmonicity of sections of sphere bundles we consider the energy functional on the space of sections of a sphere bundle over a riemannian manifold m langle cdot cdot rangle equipped with the sasaki metric and discuss the characterising condition for critical points furthermore we provide a useful method for computing the tension field in some particular situations such a method is shown to be adequate for many tensor fields defined on manifolds m equipped with a g structure compatible with langle cdot cdot rangle this leads to the construction of several new examples of differential forms which are harmonic sections or determine a harmonic map from m langle cdot cdot rangle into its sphere bundle

plurisubharmonic approximation and boundary values of plurisubharmonic functions we study the problem of approximating plurisubharmonic functions on a bounded domain omega by continuous plurisubharmonic functions defined on neighborhoods of omega over bar it turns out that t

short time behaviour of a modified laplacian coflow of g2 structures abstract we modify the laplacian coflow of co closed g 2 structures d d t where is the closed dual 4 form of a g 2 structure the modified flow is now weakly parabolic in the direction of closed forms up to diffeomorphisms we then prove short time existence and uniqueness of solutions to the modified flow

reduction of hkt structures kt geometry is the geometry of a hermitian connection whose torsion is a 3 form hkt geometry is the geometry of a hyper hermitian connection whose torsion is a 3 form we identify nontrivial conditions for a reduction theory for these types of geometry

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

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

special metrics and triality abstract we investigate a new 8 dimensional riemannian geometry defined by a generic closed and coclosed 3 form with stabiliser psu 3 and which arises as a critical point of hitchin s variational principle we give a riemannian characterisation of this structure in terms of invariant spinor valued 1 forms which are harmonic with respect to the twisted dirac operator on 1 we establish various obstructions to the existence of topological reductions to psu 3 for compact manifolds we also give sufficient conditions for topological psu 3 structures that can be lifted to topological su 3 structures we also construct the first known compact example of an integrable non symmetric psu 3 structure in the same vein we give a new riemannian characterisation for topological quaternionic kahler structures which are defined by an sp 1 sp 2 invariant self dual 4 form again we show that this form is closed if and only if the corresponding spinor valued 1 form is harmonic for and that these equivalent conditions produce constraints on the ricci tensor

a uniform estimate for general quaternionic calabi problem with appendix by daniel barlet abstract we prove a c 0 a priori estimate on a solution of the quaternionic calabi problem on an arbitrary compact connected hkt manifold this generalizes earlier results 9 and 7 where this result was proven under certain extra assumptions on the manifold

superconformal symmetry and hyperkahler manifolds with torsion the geometry arising from michelson strominger s study of 4b supersymmetric quantum mechanics with superconformal d 2 1 symmetry is a hyperkahler manifold with torsion hkt together with a special homothety it is shown that different parameters are related via changes in potentials for the hkt target spaces for 0 1 we describe how each such hkt manifold m 4m is derived from a space n 4m 4 which is quaternionic kahler with torsion and carries an abelian instanton

conformally parallel g2 structures on a class of solvmanifolds starting from a 6 dimensional nilpotent lie group n endowed with an invariant su 3 structure we construct a homogeneous conformally parallel g 2 metric on an associated solvmanifold we classify all half flat su 3 structures that endow the rank one solvable extension of n with a conformally parallel g 2 structure by suitably deforming the su 3 structures obtained we are able to describe the corresponding non homogeneous ricci flat metrics with holonomy contained in g 2 in the process we also find a new metric with exceptional holonomy

connections with irreducible holonomy representations 3 irreducible berger algebras 17 3 1 real berger algebras 17 3 2 examples of berger algebras 18 3 2 1 conformal lie algebras 18 3 2 2 symmetric connections 20 3 2 3 symplectic lie algebras 21 3 2 4 complex lie algebras with h 6 0 23 3 3 irreducible complex berger algebras 25 3 4 simple complex berger algebras 29 3 5 complex tensor representations 39

generalised g 2 manifolds we define new riemannian structures on 7 manifolds by a differential form of mixed degree which is the critical point of a possibly constrained variational problem over a fixed cohomology class the unconstrained critical points generalise the notion of a manifold of holonomy g 2 while the constrained ones give rise to a new geometry without a classical counterpart we characterise these structures by means of spinors and show the integrability conditions to be equivalent to the supersymmetry equations on spinors in type ii supergravity theory with bosonic background fields in particular this geometry can be described by two linear metric connections with skew symmetric torsion finally we construct explicit examples by introducing the device of t duality

g2 manifolds and associative submanifolds via semi fano 3 folds we construct many new topological types of compact g 2 manifolds i e riemannian 7 manifolds with holonomy group g 2 to achieve this we extend the twisted connected sum construction first developed by kovalev and apply it to the large class of asymptotically cylindrical calabi yau 3 folds built from semi fano 3 folds constructed previously by the authors in many cases we determine the diffeomorphism type of the underlying smooth 7 manifolds completely we find that many 2 connected 7 manifolds can be realised as twisted connected sums in a variety of ways raising questions about the global structure of the moduli space of g 2 metrics many of the g 2 manifolds we construct contain compact rigid associative 3 folds which play an important role in the the higher dimensional enumerative geometry gauge theory calibrated submanifolds approach to defining deformation invariants of g 2 metrics by varying the semi fanos used to build different g 2 metrics on the same 7 manifold we can change the number of rigid associative 3 folds we produce

einstein locally conformal calibrated g_2 structures we study locally conformal calibrated g 2 structures whose underlying riemannian metric is einstein showing that in the compact case the scalar curvature cannot be positive as a consequence a compact homogeneous 7 manifold cannot admit an invariant einstein locally conformal calibrated g 2 structure unless the underlying metric is flat in contrast to the compact case we provide a non compact example of homogeneous manifold endowed with a locally conformal calibrated g 2 structure whose associated riemannian metric is einstein and non ricci flat the homogeneous einstein metric is a rank one extension of a ricci soliton on the 3 dimensional complex heisenberg group endowed with a left invariant coupled mathrm su 3 structure omega psi i e such that d omega c mathrm re psi with c in mathbb r 0 nilpotent lie algebras admitting a coupled mathrm su 3 structure are also classified