The Stability of Abstract Boundary Essential Singularities

The abstract boundary has, in recent years, proved a general and flexible way to define the singularities of space-time. In this approach an essential singularity is a non-regular boundary point of an embedding which is accessible by a chosen family of curves within finite parameter distance. Ashley and Scott proved the first theorem relating essential singularities in strongly causal space-times to causal geodesic incompleteness. Linking this with the work of Beem on the C ^r -stability of geodesic incompleteness allows proof of the stability of these singularities. Here I present this result stating the conditions under which essential singularities are C ¹-stable against perturbations of the metric.     

The Geodesic Ray Transform on Riemannian Surfaces with Conjugate Points

We study the geodesic X-ray transform X on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and, therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of X and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but it is still ill-posed if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.     

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采用流体模型理论推导了等熵平衡条件下环向转动托卡马克等离子体中带状流的色散关系。从理论上分析了环向转动对测地声模、低频带状流和声波的频率、压力和密度扰动量的影响。结果表明,环向转动对低频带状流的频率没有影响,但会使测地声模的频率逐渐增大。此外,存在环向转动时,低频带状流会具有驻波形式的压力和密度扰动量,且测地声模和声波可以沿着极向传播。而且还发现,等熵平衡可以看成是等温平衡的一种特殊情况。     

托卡马克等熵平衡条件下等离子体环向转动对带状流的影响

采用流体模型理论推导了等熵平衡条件下环向转动托卡马克等离子体中带状流的色散关系。从理论上分析了环向转动对测地声模、低频带状流和声波的频率、压力和密度扰动量的影响。结果表明,环向转动对低频带状流的频率没有影响,但会使测地声模的频率逐渐增大。此外,存在环向转动时,低频带状流会具有驻波形式的压力和密度扰动量,且测地声模和声波可以沿着极向传播。而且还发现,等熵平衡可以看成是等温平衡的一种特殊情况。     

Gaussian Thermostats as Geodesic Flows of Nonsymmetric Linear Connections

We establish that Gaussian thermostats are geodesic flows of special metric connections. We give sufficient conditions for hyperbolicity of geodesic flows of metric connections in terms of their curvature and torsion. Reproduction of the entire article for non-commercial purposes is permitted without charge.     

Double Bracket Equations and Geodesic Flows on Symmetric Spaces

In this paper we consider the geometry of Hamiltonian flows on the cotangent bundle of coadjoint orbits of compact Lie groups and on symmetric spaces. A key idea here is the use of the normal metric to define the kinetic energy. This leads to Hamiltonian flows of the double bracket type. We analyze the integrability of geodesic flows according to the method of Thimm. We obtain via the double bracket formalism a quite explicit form of the relevant commuting flows and a correspondingly transparent proof of involutivity. We demonstrate for example integrability of the geodesic flow on the real and complex Grassmannians. We also consider right invariant systems and the generalized rigid body equations in this setting. Received:23 July 1996 / Accepted: 16 December 1996     

Zonal flows in tokamak plasmas with toroidal rotation

Zonal flows in tokamak plasmas with toroidal rotation are theoretically investigated. It is found that the low-frequency branch of zonal flows, which is linearly stable in a nonrotating system, becomes linearly unstable in a rotating tokamak, and that the high-frequency branch of zonal flows, the geodesic acoustic mode, can propagate in the poloidal direction with the frequency significantly lower than the frequency of the standing wave geodesic acoustic mode in the nonrotating system.     

Singularities of Euler Flow? Not Out of the Blue!

Does three-dimensional incompressible Euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time? This blowup problem is still open. After briefly reviewing what is known and pointing out some of the difficulties, we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations. We present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup.     

Two-dimensional superintegrable metrics with one linear and one cubic integral

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta.     

Invariant characterization of Liouville metrics and polynomial integrals

In this paper a criterion for a metric on a surface to be Liouville is established, and it is given in terms of differential invariants of the metric. Moreover, here we completely solve in invariant terms the local mobility problem of a 2D metric, considered by Darboux: How many quadratic in momenta integrals the geodesic flow of a given metric possesses? The method is also applied to recognition of higher degree polynomial integrals of geodesic flows.     

Distribution of Periods of Closed Trajectories in Exponentially Shrinking Intervals

For hyperbolic flows over basic sets we study the asymptotic of the number of closed trajectories γ with periods T _γ lying in exponentially shrinking intervals ${(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.}${(x - e^{-\delta x}, x + e^{-\delta x}), \; \delta > 0, \; x \to + \infty.} A general result is established which concerns hyperbolic flows admitting symbolic models whose corresponding Ruelle transfer operators satisfy some spectral estimates. This result applies to a variety of hyperbolic flows on basic sets, in particular to geodesic flows on manifolds of constant negative curvature and to open billiard flows.     

A Holomorphic Point of View about Geodesic Completeness

Abstract

We propose to apply the idea of analytical continuation in the complex domain to the problem of geodesic completeness. We shall analyse rather in detail the cases of analytical warped products of real lines, these ones in parallel with their complex counterparts, and of Clifton-Pohl torus, to show that our definition sheds a bit of new light on the behaviour of ’singularities’ of geodesics in space-time. We also show that some geodesics, which ’end’ at finite time in the classical sense, can be naturally continued besides their ends. As a matter of fact, complex metrics naturally show a meromorphic behaviour, or a degenerating one, so we shall study also this fact in detail.     

Energy Transfer in a Fast-Slow Hamiltonian System

We consider a finite region of a lattice of weakly interacting geodesic flows on manifolds of negative curvature and we show that, when rescaling the interactions and the time appropriately, the energies of the flows evolve according to a nonlinear diffusion equation. This is a first step toward the derivation of macroscopic equations from a Hamiltonian microscopic dynamics in the case of weakly coupled systems.     

Properties of geodesic acoustic modes and the relation to density fluctuations

The geodesic acoustic mode (GAM) is a high frequency branch of zonal flows, which is observed in toroidal plasmas. Because of toroidal curvature effects, density fluctuations are excited, which are investigated with the O-mode correlation reflectometer at TEXTOR. This Letter reports on the poloidal distribution of GAM induced density fluctuation and compares them with theoretical predictions. The influence of the GAM flows on the ambient turbulence is studied, too.     

Geodesic motion on closed spaces: Two numerical examples

The geodesic structure is very closely related to the trace of the Laplace operator, involved in the calculation of the expectation value of the energy-momentum tensor in Universes with non-trivial topology. The purpose of this work is to provide concrete numerical examples of geodesic flows. Two manifolds with genus g=0 are given. In one the chaotic regions, form sets of negligible or zero measure. In the second example the geodesic flow shows the presence of measurable chaotic regions. The approach is “experimental”, numerical, and there is no attempt to an analytical calculation.     

Persistently expansive geodesic flows

We prove thatC ¹-persistently expansive geodesic flows of compact, boundaryless Riemannian manifolds have the property that the closure of the set of closed orbits is a hyperbolic set. In the case of compact surfaces we deduce that the geodesic flow isC ¹-persistently expansive if and only if it is an Anosov flow.     

Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows

This paper represents part of a program to understand the behavior of topological entropy for Anosov and geodesic flows. In this paper, we have two goals. First we obtain some regularity results forC ¹ perturbations. Second, and more importantly, we obtain explicit formulas for the derivative of topological entropy. These formulas allow us to characterize the critical points of topological entropy on the space of negatively curved metrics.Partially supported by NSF grant DMS-8514630Chaim Weizmann Research Fellow and NSF postdoctoral Research Fellow     

Singularities and conjugate points in FLRW spacetimes

Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial curvature and scale factors that near the singularity have power law behavior or power law behavior times a logarithm. When the spatial curvature is negative, the theorem is applicable to a subset of these spacetimes.     

Integrable geodesic flows and super polytropic gas equations

The polytropic gas equations are shown to be the geodesic flows with respect to an L² metric on the semidirect product space Diff(S¹)C^∞(S¹), where Diff(S¹) is the group of orientation preserving diffeomorphisms of the circle. We also show that the N=1 supersymmetric polytropic gas equation constitute an integrable geodesic flow on the extended Neveu–Schwarz space. Recently other kinds of supersymmetrizations have been studied vigorously in connection with superstring theory and are called supersymmetric-B (SUSY-B) extension. In this paper we also show that the SUSY-B extension of the polytropic gas equation form a geodesic flow on the extension of the Neveu–Schwarz space.