Killing spinors on Lorentzian manifolds

The aim of this paper is to describe some results concerning the geometry of Lorentzian manifolds admitting Killing spinors. We prove that there are imaginary Killing spinors on simply connected Lorentzian Einstein–Sasaki manifolds. In the Riemannian case, an odd-dimensional complete simply connected manifold (of dimension n≠7) is Einstein–Sasaki if and only if it admits a non-trivial Killing spinor to . The analogous result does not hold in the Lorentzian case. We give an example of a non-Einstein Lorentzian manifold admitting an imaginary Killing spinor. A Lorentzian manifold admitting a real Killing spinor is at least locally a codimension one warped product with a special warping function. The fiber of the warped product is either a Riemannian manifold with a real or imaginary Killing spinor or with a parallel spinor, or it again is a Lorentzian manifold with a real Killing spinor. Conversely, all warped products of that form admit real Killing spinors.     

Symplectic twistor spaces

The paper describes the geometry of the bundle (M, ω) of the compatible complex structures of the tangent spaces of an (almost) symplectic manifold (M, ω), from the viewpoint of general twistor spaces [3], [9], [1]. It is shown that M has an either complex or almost Kaehler twistor space iff it has a flat symplectic connection. Applications of the twistor space to the study of the differential forms of M, and to the study of mappings : N → M, where N is a Kaehler manifold are indicated.     

Singular self-dual Zollfrei metrics and twistor correspondence

We construct examples of singular self-dual Zollfrei metrics explicitly, by patching a pair of Petean’s self-dual split-signature metrics. We prove that there is a natural one-to-one correspondence between these singular metrics and a certain set of embeddings of RP³$\mathbb{R}{\mathbb{P}}^3$ to CP³$\mathbb{C}{\mathbb{P}}^3$ which has one singular point. This embedding corresponds to an odd function on R$\mathbb{R}$ that is rapidly decreasing and pure imaginary valued. The one-to-one correspondence is explicitly given by using the Radon transform.     

Homogeneous structures on three-dimensional Lorentzian manifolds

We prove that any non-symmetric three-dimensional homogeneous Lorentzian manifold is isometric to a Lie group equipped with a left-invariant Lorentzian metric. We then classify all three-dimensional homogeneous Lorentzian manifolds.     

On supersymmetric Lorentzian Einstein-Weyl spaces

We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to supergravity theories to characterise those Lorentzian EW geometries that allow for a weighted parallel spinor, calling the resulting geometries supersymmetric. The overall result is that they are either conformally related to ordinary geometries admitting parallel spinors (w.r.t. the Levi-Cività connection), or they are conformally related to certain Kundt spacetime. A full characterisation is obtained for the 4- and 6-dimensional cases.     

On the solvability of the transvection group of extrinsic symplectic symmetric spaces

Let M$M$ be a symplectic symmetric space, and let ?:M→V$?:M\to V$ be an extrinsic symplectic symmetric immersion in the sense of Krantz and Schwachhöfer (2010) [7], i.e., (V,Ω)$(V,\Omega )$ is a symplectic vector space and ?$?$ is an injective symplectic immersion such that for each point p∈M$p\in M$, the geodesic symmetry in p$p$ is compatible with the reflection in the affine normal space at ?(p)$?\left(p\right)$.     

New examples of indefinite hyper-Kähler symmetric spaces

Following the approach to pseudo-Riemannian symmetric spaces developed in [I. Kath, M. Olbrich, On the structure of pseudo-Riemannian symmetric spaces, arXiv:math.DG/0408249, 2004] we exhibit examples of indefinite hyper-Kähler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kähler symmetric spaces whose metric has signature (4,4n)$(4,4n)$. Such spaces exist if and only if n∈{0,1,3}$n\in \{0,1,3\}$.     

Space-like hypersurfaces with positive constant r-mean curvature in Lorentzian product spaces

In this paper we obtain a height estimate concerning compact space-like hypersurfaces Σ ⁿ immersed with some positive constant r-mean curvature into an (n + 1)-dimensional Lorentzian product space , and whose boundary is contained into a slice {t} × M ⁿ . By considering the hyperbolic caps of the Lorentz–Minkowski space , we show that our estimate is sharp. Furthermore, we apply this estimate to study the complete space-like hypersurfaces immersed with some positive constant r-mean curvature into a Lorentzian product space. For instance, when the ambient space–time is spatially closed, we show that such hypersurfaces must satisfy the topological property of having more than one end which constitutes a necessary condition for their existence.     

Geometrical aspects of Schlesinger''s equation

The equation (Schlesinger's equation) for the isomonodromic deformations of an (SL (2, C) connection with four simple poles on the projective line is shown to describe a holomorphic projective structure on a surface. The space of geodesics of this structure is, by a primitive version of twistor theory, a two-dimensional complex Poisson manifold containing complete rational curves. The Poisson structure degenerates on a divisor and it is shown that the complement of the divisor is a symplectic manifold which can be identified with the quotient of the moduli space of representations of a free group on three generators in SL (2, ) by the action of a braid group.     

A variational principle for actions on symmetric symplectic spaces

We present a definition of generating functions of canonical relations, which are real functions on symmetric symplectic spaces, discussing some conditions for the presence of caustics. We show how the actions compose by a neat geometrical formula and are connected to the hamiltonians via a geometrically simple variational principle which determines the classical trajectories, discussing the temporal evolution of such “extended hamiltonians” in terms of Hamilton–Jacobi-type equations. Simplest spaces are treated explicitly.     

Supergravity-induced interactions on thick branes

The gravity coupling of the symmetric space sigma model is studied in the solvable Lie algebra parametrization. The corresponding Einstein equations are derived and the energy-momentum tensor is calculated. The results are used to derive the dynamical equations of the warped five-dimensional （5D） geometry for localized bulk scalar interactions in the framework of thick brane world models. The Einstein and scalar field equations are derived for flat brane geometry in the context of minimal and non-minimal gravity-bulk scalar couplings.     

基于洛伦兹函数微扰的磁光光纤布喇格光栅线性透射谱和非线性双稳特性研究

基于非线性耦合模理论,利用逆向递推龙格-库塔法,数值研究了在磁光耦合系数中引入洛伦兹函数微扰后对磁光光纤布喇格光栅线性透射谱和非线性双稳特性的影响.研究结果表明:给磁光耦合系数引入洛伦兹微扰能在线性透射谱阻带中打开线宽极窄的一个透射窗口,微扰宽度和微扰中心位置可以影响透射窗口的位置、宽度以及峰值大小;当微扰宽度和微扰中心位置发生变化时,光栅的双稳特性表现出明显的差异,合理地选择微扰参量可以实现对其双稳特性的优化.     

Singular Hartree equation in fractional perturbed Sobolev spaces

We establish the local and global theory for the Cauchy problem of the singular Hartree equation in three dimensions, that is, the modification of the non-linear Schrödinger equation with Hartree non-linearity, where the linear part is now given by the Hamiltonian of point interaction. The latter is a singular, self-adjoint perturbation of the free Laplacian, modelling a contact interaction at a fixed point. The resulting non-linear equation is the typical effective equation for the dynamics of condensed Bose gases with fixed point-like impurities. We control the local solution theory in the perturbed Sobolev spaces of fractional order between the mass space and the operator domain. We then control the global solution theory both in the mass and in the energy space.     

Effect of high foreign-gas pressures on perpendicular and hybrid IR absorption bands for symmetric top molecules

The effect of high foreign-gas pressure (He, Ar, N₂ up to 220 atm and CO₂ up to 90 atm) on the contours of perpendicular and hybrid IR absorption bands for symmetric top molecules has been studied. It has been established that over the entire studied pressure range, the shape of the contours of the perpendicular vibrational bands ν₄ and ν₅ for chloroform, deuterochloroform, and bromoform is stable and changes relatively little as the pressure increases. The effect of collisions on the band contours is estimated by comparing the contours for mixtures of compressed gases with the contours calculated in the approximation of the J-and M-diffusion models. It has been shown that intramolecular perturbation of free rotation by Coriolis forces is the determining factor in formation of the contours of the perpendicular bands, successfully competing with the perturbing effect of the foreign gases, evidence for which comes from the insignificant changes in the shape of the bands as the pressure of the foreign gases increases. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 74, No. 1, pp. 25–30, January–February, 2007.