dimensionally dependent tensor identities by double antisymmetrization some years ago lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure and could all be considered consequences in n dimensional space of a pair of fundamental identities involving trace free p p forms where 2p n we generalize lovelock s results and by using the fact that associated with any tensor in n dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over n 1 indices we establish a very general master identity for all trace free k l forms we then show how various other special identities are direct and simple consequences of this master identity in particular we give direct application to maxwell lanczos ricci bel and bel robinson tensors and also demonstrate how relationships between scalar invariants of the riemann tensor can be investigated in a systematic manner

on the geometry of sasakian einstein 5 manifolds on simply connected five manifolds sasakian einstein metrics coincide with riemannian metrics admitting real killing spinors which are of great interest as models of near horizon geometry for three brane solutions in superstring theory 24 we expand on the recent work of demailly and kollar 14 and johnson and kollar 20 who give methods for constructing kahler einstein metrics on log del pezzo surfaces by a previous result of the first two authors 9 circle v bundles over log del pezzo surfaces with kahler einstein metrics have sasakian einstein metrics on the total space of the bundle here these simply connected 5 manifolds arise as links of isolated hypersurface singularities which by the well known work of smale 36 together with 11 must be diffeomorphic to s 5 l s 2 s 3 more precisely using methods from mori theory in algebraic geometry we prove the existence of 14 inequivalent sasakian einstein structures on s 2 s 3 and infinite families of such structures on l s 2 s 3 with 2 l 7 we also discuss the moduli problem for these sasakian einstein structures

involutive systems of differential equations einstein s strength versus cartan s degre d arbitraire three new theorems relating einstein s notions of strength and compatibility to the field of the initial value problem are presented these theorems result i in a first proof of matthews conjectures concerning this relation for a wider class of systems of partial differential equations ii in a new interpretation of einstein s compatibility condition and iii in the exact relation between einstein s strength and cartan s degre d arbitraire

einstein metrics on s 2 bundles new einstein metrics are constructed on the associated bbb r bbb p 2 s 2 and bbb r 2 bundles of principal circle bundles with base a product of kahler einstein manifolds with positive first chern class and with euler class a rational linear combination of the first chern classes these einstein metrics represent different generalizations of the well known einstein metrics found by berard bergery d page c pope n koiso and y sakane corresponding new einstein weyl structures are also constructed

a lanczos potential in kerr geometry a lanczos potential is a tensor of rank 3 for which a certain combination of derivatives always equals the weyl tensor general existence theorems have been proved before and some explicit expressions for lanczos potentials in certain spacetimes have been found in this paper we explicitly study a lanczos potential in kerr spacetime obtained from a previously studied flat connection as an application it is shown that the mass can be expressed as an integral of the lanczos potential

the weyl lanczos equations and the lanczos wave equation in four dimensions as systems in involution the weyl lanczos equations in four dimensions form a system in involution we compute its cartan characters explicitly and use janet riquier theory to confirm the results in the case of all space times with a diagonal metric tensor and for the plane wave limit of space times we write the lanczos wave equation as an exterior differential system and with assistance from janet riquier theory we compute its cartan characters and find that it forms a system in involution we compare these cartan characters with those of the weyl lanczos equations all results hold for the real analytic case

involution and constrained dynamics ii the faddeev jackiw approach for pt i see seiler et al ibid vol 28 p 4431 1995 we study the symplectic approach to first order systems with constraints from the point of view of the formal theory of differential equations we concentrate especially on systems without first class constraints and give a geometric interpretation of an approach recently proposed by barcelos neto and wotzasek 1992 we further study the numerical properties of this approach we also comment on some problems concerning the application to field theories

on the location of resonances for schrodinger operators in the semiclassical limit i resonances free domains abstract we show that under suitable non trapping conditions on the potential v at energy e the schrodinger operator h h 2 v exhibits no resonance in a neighbourhood of e for small h we briefly discuss some critical values of e like thresholds or barrier tops

locally conformally flat quasi einstein manifolds in this paper we prove that any complete locally conformally flat quasi einstein man ifold of dimension n 3 is locally a warped product with n 1 dimensional fibers of constant curvature this result includes also the case of locally conformally flat gradient ricci solitons

metrics with prescribed ricci curvature near the boundary of a manifold suppose m is a manifold with boundary choose a point o in partial m we investigate the prescribed ricci curvature equation mathop mathrm ric nolimits g t in a neighborhood of o under natural boundary conditions the unknown g here is a riemannian metric the letter t on the right hand side denotes a 0 2 tensor our main theorems address the questions of the existence and the uniqueness of solutions we explain among other things how these theorems may be used to study rotationally symmetric metrics near the boundary of a solid torus mathcal t the paper concludes with a brief discussion of the einstein equation on mathcal t

shape resonances a comparison between parabolic and flat bottom potentials abstract sharp exponential bounds are obtained for resonance states and complex eigenvalues z for the schrodinger equation h d 2 dx 2 v z on 0 in a compact region v x is either a square well or a parabolic potential as x v x v v m x for some 0 the three cases v v 0 and 0 v v m are considered results are used to study the stratified wave propagator k c 2 y x y depending on the shape of the velocity function c s resonances of k are shown to exist

on the strength of maxwell s equations the strength of a set of field equations first defined by einstein as the number of taylor coefficients of field variables that could be chosen arbitrarily is used to show that the amount of initial data required by the electromagnetic formulation of maxwell s theory in free space is equal without approximation to that required by the potential formulation in each formulation the strength is interpreted in terms of the amount of initial data required to provide a solution of the cauchy initial value problem and in terms of the invariance properties of the formulation equality of the strengths of the two formulations of maxwell s theory is used to support the assertion that knowledge of the strengths of other established field theories provides a means for predicting the possible existence of unknown formulations of the theories

involution and constrained dynamics i the dirac approach we study the theory of systems with constraints from the point of view of the formal theory of partial differential equations for finite dimensional systems we show that the dirac algorithm completes the equations of motion to an involutive system we discuss the implications of this identification for field theories and argue that the involution analysis is more general and flexible than the dirac approach we also derive intrinsic expressions for the number of degrees of freedom

spectral resonances which become eigenvalues the stationary schrodinger equation is x 2 v x z for 2 r dx if the potential is bounded below singular only atx 0 negative on some compact interval and behaves likev x 1 x asx with 2 0 then the system admits shape resonances which continuously become eigenvalues as increases here 0 and for 2 a sufficiently large is required exponential bounds are obtained on im z as approaches a threshold the group velocity near threshold is also estimated

the lanczos potential for the weyl curvature tensor existence wave equation and algorithms in the last few years renewed interest in the 3 tensor potential l abc proposed by lanczos for the weyl curvature tensor has not only clarified and corrected lanczos9s original work but generalized the concept in a number of ways in this paper we first of all carefully summarize and extend some aspects of these results make some minor corrections and clarify some misunderstandings in the literature the following new results are also presented the computer checked complicated second order partial differential equation for the 3 potential in arbitrary gauge for weyl candidates satisfying bianchi type equations is given in those n dimensional spaces with arbitrary signature for which the potential exists this is easily specialized to lanczos potentials for the weyl curvature tensor it is found that it is only in four dimensional spaces with arbitrary signature that the nonlinear terms disappear and that certain awkward second order derivative terms cancel for four dimensional spacetimes with lorentz signature this remarkably simple form was originally found by illge using spinor methods it is also shown that for most four dimensional vacuum spacetimes any 3 potential in the lanczos gauges which satisfies a simple homogeneous wave equation must be a lanczos potential for the non zero weyl curvature tensor of the background vacuum spacetime this result is used to prove that the form of a possible lanczos potential recently proposed by dolan and kim for a class of vacuum spacetimes is in fact a genuine lanczos potential for these spacetimes

on conserved quantities of certain cohomogeneity one ricci flat equations we give a classification of continuously differentiable on shell integrals which are linear or quadratic in momenta for a hamiltonian system with constraint associated to the ricci flat cohomogeneity one einstein equations for two special classes of principal orbits hypersurfaces relations to known on shell integrals and to painleve analysis are discussed

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

propagation estimates for dispersive wave equations application to the stratified wave equation the plane stratified wave equation t2 h 0 with h c y 2 z2 is studied where z x y x rk y r1 and c y c 0 as y solutions to such an equation are solved for the propagation of waves through a layered medium and can include waves which propagate in the x directions only i e trapped modes this leads to a consideration of the pseudo differential wave equation t2 x 0 such that the dispersion relation 2 is analytic and satisfies c1 2 c2 for c 0 uniform propagation estimates like x t e utp 0 dkx c 1 t e 0 dkx are obtained where ut is the evolution group p are projection operators onto the hilbert space of initial conditions h and e is the local energy density in special cases scattering of trapped modes off a local perturbation satisfies the causality estimate p jsp k c for each 1 2 here p j p k are remote outgoing detector incoming transmitter projections for the jth kth trapped mode also r is compact so the projections lo

exterior differential systems janet riquier theory and the riemann lanczos problems in two three and four dimensions we discuss the riemann lanczos problems in two three and four dimensions using the theory of exterior differential systems and janet riquier theory we show that the riemann lanczos problem in two dimensions is always a system in involution for each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result in three dimensions the riemann lanczos problem is not in involution as an identity occurs this does not prevent the existence of singular solutions and we give an example for the reduced godel space time a prolongation of this problem whereby an integrability condition is added leads to a prolonged system in involution the riemann lanczos problem in four dimensions is not in involution and needs to be prolongated as bampi and caviglia suggested but singular solutions of it can be found and we give examples for the godel kasner and debever hubaut space times

the identities of the algebraic invariants of the four dimensional riemann tensor this paper extends the investigation of the invariants of the riemann tensor to include the invariants that are of odd degree in the trace free ricci tensor it is shown that these invariants can be expressed in terms of 15 such invariants that are irreducible as a consequence it is possible to write down a complete set of invariants of the riemann tensor several syzygies for these invariants have been found and examples of these are given these syzygies suggest there may be several new syzygies of invariants with even degree in the trace free ricci tensor a large number of these have also been found and are discussed in the paper

linear independence of renormalization counterterms in curved space times of arbitrary dimensionality the counterterms in the lagrangian of a renormalizable quantum field theory that involve the riemann curvature tensor are considered it is proved that for six or fewer dimensions the counterterms not containing derivatives of the riemann tensor are linearly independent it is shown that there appears to be a maximum space time dimension for which identities can exist among invariants involving the product of n riemann tensors without derivatives acting on them space times that would require these products as counterterms for a renormalizable theory have a dimensionality which is one higher than this maximum dimension this makes it plausible that the required set of counterterms not involving derivatives of the riemann tensor is linearly independent in arbitrary dimensions in the appendix a relation cubic in the weyl tensor which relates invariants cubic in the riemann tensor in five dimensions is proved

the wave equation for the lanczos potential i the non local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor c abcd of weyl for this field lanczos found a tensor potential l abc with 16 independent components we can make l abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions l smallmatrix ab s s endsmallmatrix 0 both fields c abcd l abc satisfy wave equations the wave equation satisfied by l abc is nonlinear even in vacuo however a linear spinor wave equation for the lanczos potential has been found by illge but no correct tensor wave equation for l abc has yet been published here we derive a correct tensor wave equation for l abc and when it is simplified with the aid of some four dimensional identities it is equivalent to illge s wave equation we also show that the nonlinear spinor wave equation of penrose for the weyl field can be derived from illge s spinor wave equation a set of analogues of well known results of classical electromagentic radiation theory can now be given we indicate how a green s function approach to gravitational radiation could be based on our tensor wave equation when a global study of space time is attempted

involution analysis of the partial differential equations characterizing hamiltonian vector fields in a recent article certain underdetermined linear systems of partial differential equations connected with lie poisson structures have been studied they were constructed via power series solutions of the evolution equation for a given hamiltonian we extend the results to arbitrary poisson manifolds correct an error in the case of degenerate poisson structures and show that these linear systems simply characterize hamiltonian vector fields our basic tool is the formal theory of differential equations with its central concept of an involutive system

differential invariant algebras of lie pseudo groups abstract the aim of this paper is to describe in as much detail as possible and constructively the structure of the algebra of differential invariants of a lie pseudo group acting on the submanifolds of an analytic manifold under the assumption of local freeness of a suitably high order prolongation of the pseudo group action we develop computational algorithms for locating a finite generating set of differential invariants a complete system of recurrence relations for the differentiated invariants and a finite system of generating differential syzygies among the generating differential invariants in particular if the pseudo group acts transitively on the base manifold then the algebra of differential invariants is shown to form a rational differential algebra with non commuting derivations the essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules the usual symbol module associated with the infinitesimal determining system of the pseudo group and a new prolonged symbol module constructed from the symbols of the annihilators of the prolonged pseudo group generators modulo low order complications the generating differential invariants and differential syzygies are in one to one correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module our algorithms and proofs are all constructive and rely on combining the moving frame approach developed in earlier papers with grobner basis algorithms from commutative algebra

the weyl lanczos relations and the four dimensional lanczos tensor wave equation and some symmetry generators we examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the cartan theory of exterior differential systems to the weyl lanczos equations and to the lanczos wave equation in four dimensions we look at a number of examples of symmetries for the weyl lanczos equations in four dimensions and give examples of isovectors when the solution manifold is the schwarzschild kasner or godel space time solutions of the weyl lanczos system are automatically solutions of the lanczos wave equation we give examples of symmetry generators for the lanczos wave equation and find that they are not automatically symmetry generators for the weyl lanczos equations

the riemann lanczos equations in general relativity and their integrability the aim of this paper is to examine the riemann lanczos equations and how they can be made integrable they consist of a system of linear first order partial differential equations that arise in general relativity whereby the riemann curvature tensor is generated by an unknown third order tensor potential field called the lanczos tensor our approach is based on the theory of jet bundles where all field variables and all their partial derivatives of all relevant orders are treated as independent variables alongside the local manifold coordinates xa on the given space time manifold m this approach is adopted in a cartan s method of exterior differential systems b vessiot s dual method using vector field systems and c the janet riquier theory of systems of partial differential equations all three methods allow for the most general situations under which integrability conditions can be found they give equivalent results namely that involutivity is always achieved at all generic points of the

kahler einstein metrics on complex surfaces with c 1 0 various estimates of the lower bound of the holomorphic invariant m defined in t are given here by using branched coverings potential estimates and lelong numbers of positive d closed 1 1 currents of certain type etc these estimates are then applied to produce kahler einstein metrics on complex surfaces withc1 0 in particular we prove that there are kahler einstein structures withc1 0 on any manifold of differential type cp 2 overline ncp 2 3 leqq n leqq 8

a gap theorem for positive einstein metrics on the four sphere we show that there exists a universal positive constant varepsilon 0 0 with the following property let g be a positive einstein metric on the four sphere s 4 if the yamabe constant of the conformal class g satisfies begin aligned y s 4 g frac 1 sqrt 3 y s 4 g mathbb s varepsilon 0 end aligned where g mathbb s denotes the standard round metric on s 4 then up to rescaling g is isometric to g mathbb s this is an extension of gursky s gap theorem for positive einstein metrics on s 4

on effective constraints for the riemann lanczos system of equations there have been conflicting points of view concerning the riemann lanczos problem in three and four dimensions using direct differentiation on the defining partial differential equations massa and pagani in four dimensions and edgar in dimensions n 3 have argued that there are effective constraints so that not all riemann tensors can have lanczos potentials using cartan s criteria of integrability of ideals of differential forms bampi and caviglia have argued that there are no such constraints in dimensions n 4 and that in these dimensions all riemann tensors can have lanczos potentials in this article we give a simple direct derivation of a constraint equation confirm explicitly that known exact solutions of the riemann lanczos problem satisfy it and argue that the bampi and caviglia conclusion must therefore be flawed in support of this we refer to the recent work of dolan and gerber on the three dimensional problem by a method closely related to that of bampi and caviglia they have fou

trapped rays in cylindrically symmetric media and poles of the analytically continued resolvent abstract the acoustic stratified propagator is defined as the positive self adjoint operatork c2 y y z 1 y zon l 2 rn c2 1dnz wherez x y x rk y rm bounds on the weighted resolvent ofk defined as z k 1 z for 1 2 and z 1 z 2 1 2 are obtained at high energies i e in the semi classical limit re 0 using the virial conditionc y c ec 0 this implies energy decay estimates for trapped waves in fiber optics with further smoothness assumptions oncand the resolvent can be analytically extended onto the second riemann sheet the failure of the virial condition on some radial interval form 2 implies the existence ofresonant traveling modesfor the wave equation t2 k 0 in particular it is shown that ifn m 2 thenkhas resonance states at high energies no such states exist ifm 1 this is explained using a shape resonance picture

stark ladder resonances for small electric fields we prove the existence of resonances in the semi classical regime of smallh for stark ladder hamiltonians h h f equiv h 2 frac d 2 dx 2 v fx in one dimension the potentialv is a real periodic function with period which is the restriction to of a function analytic in a strip about the electric field strengthf satisfies the bounds v f 0 in general the imaginary part of the resonances are bounded above by open image in new window for some 0 0 independent ofh andf forh sufficiently small this is an extension of the oppenheimer formula for the stark effect to the case of periodic potentials

identities of the scalars of the four dimensional riemannian manifold a general algorithm is presented for the calculation of identities relating scalars constructed from appropriate contractions of the riemann tensor ricci tensor and curvature scalar the algorithm is applied to determining the identities among the quartic scalars

old and new results for superenergy tensors from dimensionally dependent tensor identities it is known that some results for spinors and in particular for superenergy spinors are much less transparent and require a lot more effort to establish when considered from the tensor viewpoint in this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4 dimensional identities by straightforward tensor methods in a signature independent manner in particular we consider the quadratic identity for the bel robinson tensor tabcxtabcy xy tabcdtabcd 4 and also the new conservation law for the chevreton tensor both of which have been obtained by spinor means both of these results are rederived by tensor means for 4 dimensional spaces of any signature using dimensionally dependent identities and moreover we are able to conclude that there are no direct higher dimensional analogs in addition we demonstrate a simple way to show the nonexistense of such identities via counter examples in particular we show that there is no nontrivial bel ten

generalized kronecker and permanent deltas their spinor and tensor equivalents and applications the aim of this paper is fourfold i to introduce a generalized permanent delta on an equal footing with the generalized kronecker delta to use for the symmetries of any tensor or spinor ii to cite an ancillary reference source of comprehensive tensorial and spinorial combinatorial formulas for both iii to table spinor equivalents of these individual tensors and give examples of their usage and iv to tabulate the tensor equivalents of various useful combinations of their spinor forms

on the arbitrariness of the general solution of an involutive partial differential equation the relationship between the strength of a differential equation as introduced by einstein its cartan characters and its hilbert polynomial is studied using the framework of formal theory previous results are extended to nonlinear equations of arbitrary order and to overdetermined systems the problem of computing the number of arbitrary functions in the general solution is treated finally the effect of gauge symmetries is considered

some solutions of the lanczos vacuum wave equation in general relativity the non local part of the gravitational field is described by the 10 degrees of freedom of the weyl conformal curvature tensor c abcd in every space time the weyl field c abcd is derivable from a potential l abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions l smallmatrix ab s s endsmallmatrix 0 are imposed the potential l abc discovered by lanczos was shown by illge to have an extremely simple vacuum wave equation namely square l abc equiv g sm l abc s m 0 using tensor spinor and spin coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields