Vacuum type N space-times admitting homothetic vector fields with isolated fixed points

We consider vacuum space-times (M, g) which are of Petrov type N on an open dense subset ofM, and which admit (proper) homothetic vector fields with isolated fixed points. We prove that if such is the case then, at the fixed point, (M,g) is flat and the homothetic bivector,X _[a;b] , is necessarily simple-timelike. Furthermore, we prove that if the homothetic bivector remains simple-timelike in some neighbourhood of the fixed point then, around the fixed point, the space-time in question is a pp-wave. The paper ends with a local characterization and some examples of space-tunes satisfying these conditions.     

Homothetic maps of the space-time distance function and differentiability

Let (M ₁,g ₁) and (M ₂,g ₁) be time-oriented space-times. Letd _i(p,q) be the supremum of lengths of future directed causal curves inM _i fromp toq. Ifq is not in the future ofp, thend _i (p, q)=0. A distance homothetic mapf is a function fromM ₁ ontoM ₂ which is not assumed to be continuous, but which satisfiesd ₂(f(p),f(q))=cd ₁(p,q) for allp,q M ₁. IfM ₁ is strongly causal, then the distance homothetic mapf is a diffeomorphism which mapsg ₁ to a scalar multiple ofg ₂. Thus for strongly causal space-times, distance homothetic maps are homothetic in the usual sense. WhenM ₁ is not strongly causal, distance homothetic maps are not necessarily differentiable nor even continuous. An example is given of a space-time which has discontinuous maps which are one to one, onto, and distance preserving.     

On a covariant version of Caianiello’s model

Caianiello’s derivation of Quantum Geometry through an isometric embedding of the spacetime (M, g̃) in the pseudo-Riemannian structure (T*M, g* _AB ) is reconsidered. In the new derivation, using a non-linear connection and the bundle formalism, we obtain a Lorentzian-type structure in the 4-dimensional manifold M that is covariant under arbitrary local coordinate transformations in M. We obtain that if models with maximal acceleration are non-trivial, gravity should be supplied with other interactions in a unification framework.     

Flows on Quaternionic-Kähler and Very Special Real Manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product M×N of a quaternionic-Kähler manifold M of negative scalar curvature and a very special real manifold N of dimension n0. Such gradient flows are generated by the ``energy function' f=P<sup>2</sup>, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kähler manifolds. For the homogeneous quaternionic-Kähler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point pM such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kähler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kähler manifolds we find degenerate local minima. This work was supported by the priority programme ``String Theory'of the Deutsche Forschungsgemeinschaft.     

On conformal Killing 2-form of the electromagnetic field

Let D:C^∞Λ^pM→C^∞(T^*MΛ^pM) be the first order linear differential operator on an n-dimensional (1≤p≤n−1) pseudo-Riemannian manifold (M,g). We have by the representation theory of orthogonal group, that the tangent bundle of this operation decomposes into the orthogonal and irreducible sum of forms of degree p+1 (which gives the exterior differential d), the forms of degree p−1 (defining the codifferential d^*) and the trace-free part of the partial symmetrization (the corresponding first order operator is denoted by D). The general forms in the kernel of D are closely related to conformal Killing vector fields, called conformal Killing p-forms, while those in kernel of d are called closed conformal Killing p-forms or, according to another terminology, planar p-forms. In particular an arbitrary planar 1-form ω is dual (by g) to the special concircular vector field ξ. We consider some local properties for the closed conformal Killing p-forms. As an application we present examples of decomposition into irreducible components for the electromagnetic field 2-form ω and its covariant derivative in four-dimensional space–time. In particular, we prove that the energy–momentum tensor T of the electromagnetic field is a symmetric conformal Killing tensor if the electromagnetic field 2-form ω is a conformal Killing form.     

Conformal groups and conformally equivalent isometry groups

It is shown that if ann dimensional Riemannian or pseudo-Riemannian manifold admits a proper conformal scalar, every (local) conformal group is conformally isometric, and that if it admits a proper conformal gradient every (local) conformal group is conformally homothetic. In the Riemannian case there is always a conformal scalar unless the metric is conformally Euclidean. In the case of a Lorentzian 4-manifold it is proved that the only metrics with no conformal scalars (and hence the only ones admitting a (local) conformal group not conformally isometric) are either conformal to the plane wave metric with parallel rays or conformally Minkowskian.     

Homothetic transformations with fixed points in spacetime

A study is made of homothetic motions with fixed points in spacetime. Some general properties of such spacetimes are established, and a characterization of generalized plane wave spacetimes is proved. A general discussion of homothetic motions in Einstein's theory is given.This is in the sense that no open subset ofM is flat.     

Commuting Operators and Separation of Variables for Laplacians of Projectively Equivalent Metrics

Let two Riemannian metrics g and g on one manifold M ⁿ have the same geodesics (considered as unparameterized curves). Then we can construct invariantly n commuting differential operators of second order. The Laplacian _g of the metric g is one of these operators. For any x M ⁿ , consider the linear transformation G of T _x M ⁿ given by the tensor g ^Ig_j . If all eigenvalues of G are different at one point of the manifold then they are different at almost every point; the operators are linearly independent and their symbols are functionally independent. If all eigenvalues of G are different at each point of a closed manifold then it can be covered by the n-torus and we can globally separate the variables in the equation _g f = f on this torus.     

Homoclinic points in symplectic and volume-preserving diffeomorphisms

LetM ⁿ be a compactn-dimensional manifold and ω be a symplectic or volume form onM ⁿ. Let ? be aC ¹ diffeomorphism onM ⁿ that preserves ω and letp be a hyperbolic periodic point of Φ. We show that genericallyp has a homoclinic point, and moreover, the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp. Takens [11] obtained the same result forn=2.     

Construction of Fredholm representations and a modification of the Higson-Roe corona

The Fredholm representation theory is well adapted to the construction of homotopy invariants of non-simply-connected manifolds by means of the generalized Hirzebruch formula [σ(M)] = 〈L(M)ch _A f*ξ, [M]〉 ∈ K _A ⁰(pt) ⊗ Q, where A = C*[π] is the C*-algebra of the group π, π = π ₁(M). The bundle ξ ∈ K _A ⁰(Bπ) is the canonical A-bundle generated by the natural representation π → A. Recently, the first author constructed a natural family of Fredholm representations that lead to a symmetric vector bundle on the completion of the fundamental group with a modification of the Higson-Roe corona, provided that the completion is a closed manifold.     

The Klein-Gordon operator in conformal space

We consider the Minkowski space M₄ as a local chart of a compact differentiable pseudo-Riemannian manifold M⁴_c, on which the whole conformal group $\text{O(2, 4)}\text{Z}{}_2$ acts continuously. We investigate the conditions under which functions or differential operators on the space M₄ can be uniquely continued to the conformal manifold M⁴_c. This is done by using methods well-known in the theory of differentiable manifolds. In particular, we show that the Klein-Gordon operator □+m² can be uniquely continued to the space M⁴_c and we discuss the conformal invariance of the Klein-Gordon equation on the manifold M⁴_c.     

The R.I. Pimenov unified gravitation and electromagnetism field theory as semi-Riemannian geometry

More than forty years ago R.I. Pimenov introduced a new geometry—semi-Riemannian one—as a set of geometrical objects consistent with a fibering pr: M _n → M _m . He suggested the heuristic principle according to which the physically different quantities (meter, second, Coulomb, etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple-fibered semi-Riemannian geometry is the most appropriate one for the treatment of more than three different physical quantities as unified geometrical field theory. Semi-Euclidean geometry ³ R ₅⁴ with 1-dimensional fiber x ⁵ and 4-dimensional Minkowski space-time as a base is naturally interpreted as classical electrodynamics. Semi-Riemannian geometry ³ V ₅⁴ with the general relativity pseudo-Riemannian space-time ³ V ₄, and 1-dimensional fiber x ⁵, responsible for the electromagnetism, provides the unified field theory of gravitation and electromagnetism. Unlike Kaluza-Klein theories, where the fifth coordinate appears in nondegenerate Riemannian or pseudo-Riemannian geometry, the theory based on semi-Riemannian geometry is free from defects of the former. In particular, scalar field does not arise. The text was submitted by the author in English.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

Nuclear matter properties in relativistic mean-field model with σ- ω coupling

The σ-ω coupling is introduced phenomenologically in the linear σ-ω model to study the nuclear matter properties. It is shown that not only the effective nucleon mass M^* but also the effective σ meson mass m _σ ^* and the effective ω meson mass m _ω ^* are nucleon-density-dependent. When the model parameters are fitted to the nuclear saturation point, with the nuclear radius constant r ₀ = 1.14 fm and volume energy a ₁ = 16.0 MeV, as well as to the effective nucleon mass M ^* = 0.85M, the model yields m _σ ^* = 1.09m _σ and m _ω ^* = 0.90m _ω at the saturation point, and the nuclear incompressibility K ₀ = 501 MeV. The lowest value of K₀ given by this model by adjusting the model parameters is around 227 MeV. Received: 23 March 2001 / Accepted: 8 June 2001     

Chiral approach to nuclear matter: role of two-pion exchange with virtual delta-isobar excitation

We extend a recent three-loop calculation of nuclear matter by including the effects from two-pion exchange with single and double virtual Δ(1232)-isobar excitation. Regularization dependent short-range contributions from pion-loops are encoded in a few NN-contact coupling constants. The empirical saturation point of isospin-symmetric nuclear matter, , ρ₀=0.16 fm⁻³, can be well reproduced by adjusting the strength of a two-body term linear in density (and tuning an emerging three-body term quadratic in density). The nuclear matter compressibility comes out as K=304 MeV. The real single-particle potential U(p,k_f0) is substantially improved by the inclusion of the chiral πNΔ-dynamics: it grows now monotonically with the nucleon momentum p. The effective nucleon mass at the Fermi surface takes on a realistic value of M^*(k_f0)=0.88M. As a consequence of these features, the critical temperature of the liquid-gas phase transition gets lowered to the value T_c15 MeV. In this work we continue the complex-valued single-particle potential U(p,k_f)+iW(p,k_f) into the region above the Fermi surface p>k_f. The effects of 2π-exchange with virtual Δ-excitation on the nuclear energy density functional are also investigated. The effective nucleon mass associated with the kinetic energy density is . Furthermore, we find that the isospin properties of nuclear matter get significantly improved by including the chiral πNΔ-dynamics. Instead of bending downward above ρ₀ as in previous calculations, the energy per particle of pure neutron matter and the asymmetry energy A(k_f) now grow monotonically with density. In the density regime ρ=2ρ_n<0.2 fm⁻³ relevant for conventional nuclear physics our results agree well with sophisticated many-body calculations and (semi)-empirical values.     

Fedosov Deformation Quantization as a BRST Theory

The relationship is established between the Fedosov deformation quantization of a general symplectic manifold and the BFV-BRST quantization of constrained dynamical systems. The original symplectic manifold ℳ is presented as a second class constrained surface in the fibre bundle ?^* _ρℳ which is a certain modification of a usual cotangent bundle equipped with a natural symplectic structure. The second class system is converted into the first class one by continuation of the constraints into the extended manifold, being a direct sum of ?^* _ρℳ and the tangent bundle Tℳ. This extended manifold is equipped with a nontrivial Poisson bracket which naturally involves two basic ingredients of Fedosov geometry: the symplectic structure and the symplectic connection. The constructed first class constrained theory, being equivalent to the original symplectic manifold, is quantized through the BFV-BRST procedure. The existence theorem is proven for the quantum BRST charge and the quantum BRST invariant observables. The adjoint action of the quantum BRST charge is identified with the Abelian Fedosov connection while any observable, being proven to be a unique BRST invariant continuation for the values defined in the original symplectic manifold, is identified with the Fedosov flat section of the Weyl bundle. The Fedosov fibrewise star multiplication is thus recognized as a conventional product of the quantum BRST invariant observables. Received: 28 April 2000 / Accepted: 6 December 2000     

X-ray M4,5 resonant Raman scattering from Gd with final 4p hole: calculations with 4p–4d–4f configuration interaction in the final state and comparison with the experiment

We present calculations of resonant Raman scattering (RRS) at the M_4,5 thresholds of Gd in the scattering channel 3d¹⁰4f⁷→3d⁹4f⁸→[4p⁵4f⁸↔4d⁸4f⁹]. We have included in the final state the interaction between the two configurations within the brackets, having one 4p and two 4d holes, respectively. The influence of the configuration interaction on the scattering spectra is shown to be important. The calculations are made within a purely ionic model including only the spectral features dispersing with the incident photon energy and do not account for the M₄ to M₅ Coster-Kronig conversion. The calculations are compared with recent experimental results on Gd metal. The agreement is excellent when choosing the excitation energy in the M₅ region. In the M₄ region the calculations agree with the measurements by assuming that the Coster-Kronig contribution is approximated in shape by the RRS spectrum measured with direct M₅ excitation. The implications of the results are discussed.     

Renormalization group fixed points in the local potential approximation ford≧3

In the Local Potential Approximation, renormalization group equations reduce to a semilinear parabolic partial differential equation. Felder [8] has derived this equation and has constructed a family of non-trivial fixed pointsu _2n ^* (n=2,3,4,...) which have the form ofn-well potentials and exist in the ranges of dimensions 2<d<2+2/n–1. In this paper we show that ifd4, then these non-trivial fixed points disappear, and if 3d<4 then we have only theu ₄ ^* fixed point.Research supported by CNPq, Brazil     

Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits

Let M be a smooth compact manifold of dimension at least 2 and Diff ^r (M) be the space of C ^r smooth diffeomorphisms of M. Associate to each diffeomorphism f;isin; Diff ^r (M) the sequence P _n (f) of the number of isolated periodic points for f of period n. In this paper we exhibit an open set N in the space of diffeomorphisms Diff ^r (M) such for a Baire generic diffeomorphism f∈N the number of periodic points P _n f grows with a period n faster than any following sequence of numbers {a _n } _{n ∈ Z +} along a subsequence, i.e. P _n (f)>a _ni for some n _i →∞ with i→∞. In the cases of surface diffeomorphisms, i.e. dim M≡2, an open set N with a supergrowth of the number of periodic points is a Newhouse domain. A proof of the man result is based on the Gontchenko–Shilnikov–Turaev Theorem [GST]. A complete proof of that theorem is also presented. Received: 27 January 1999 / Accepted: 23 November 1999     

Dynamics of forest fire under rotational constraint

A rotationally constrained forest fire model is studied on square and triangular lattices of size 400×400. The critical probabilityp _c for onset of fire propagation is determined. The scaling relationsMt ^d _r, R_gt^v andMR _g ^d _f are analysed at fire propagation probabilityp=p _c whereM is the number of burnt trees,R _g the radius of gyration andd _f the fractal dimension of the cluster of burnt trees at timet. Numerical estimates ofd _t, v andd _f have been obtained.