On the isotropy subalgebras of Lie algebras of conformal vector fields

The conformal isotropy algebra of a point m in an n-manifold with a metric of arbitrary signature is shown to be locally reducible, by a conformal change of the metric, to a homothetic algebra near m iff, by choice of a chart, its constituent vector fields are simultaneously linearisable at m and, for n≥3, a necessary and sufficient condition for this in terms of the first and second derivatives of these fields at m is given. The implications for the Riemannian case and the Lorentzian case are investigated. In contrast to the former, a Lorentzian manifold admitting a conformal vector field that is not linearisable at some point need not be conformally flat. Relevant four-dimensional examples are provided.     

New cohomogeneity one metrics with Spin(7) holonomy

We construct new explicit non-singular metrics that are complete on non-compact Riemannian 8-manifolds with holonomy Spin(7). One such metric, which we denote by , is complete and non-singular on . The other complete metrics are defined on manifolds with the topology of the bundle of chiral spinors over S⁴, and are denoted by , and . The metrics on and occur in families with a non-trivial parameter. The metric on arises for a limiting value of this parameter, and locally this metric is the same as the one for . The new Spin(7) metrics are asymptotically locally conical (ALC): near infinity they approach a circle bundle with fibres of constant length over a cone whose base is the squashed Einstein metric on . We construct the covariantly constant spinor and calibrating 4-form. We also obtain an L²-normalisable harmonic 4-form for the manifold, and two such 4-forms (of opposite dualities) for the manifold.     

Growth conditions, Riemannian completeness and Lorentzian causality

The stationary-Randers correspondence (SRC) provides a deep connection between the property of standard stationary spacetimes being globally hyperbolic, and the completeness of certain Finsler metrics of Randers type defined on the fibres. In order to establish further results, we investigate pointwise conformal transformations of certain Riemannian metrics on the fibres and growth conditions on the corresponding conformal factors. In general, a conformal transformation may map a complete Riemannian metric onto a complete or incomplete metric. We prove a theorem for the growth of the conformal factor such that the conformally transformed Riemannian metric is also complete. As an application, we establish novel relations between the completeness of Riemannian metrics, growth conditions on conformal factors and the Cauchy hypersurface condition on the fibres of a standard stationary spacetime. These results also imply new conditions for the completeness of Randers-type metrics by the application of the SRC.     

On the structure of DeWitt supermanifolds

We show that for a wide and most natural class of (possibly infinite-dimensional) Grassmannian algebras of coefficients, the structure sheaf of every smooth DeWitt supermanifold is acyclic (i.e. its cohomology vanishes in positive degree). This result was previously known for finite-dimensional ground algebras and is new even for the original DeWitt algebra of supernumbers /GL∞. From here we deduce that (equivalence classes of) smooth DeWitt supermanifolds over a fixed ground algebra and of graded smooth manifolds are in a natural bijection with each other. However, contrary to what was stated previously by some authors, this correspondence fails to be functorial; so it happens, for instance, for Rogers' ground algebra B∞. Finally, we observe that every DeWitt super Lie group is a deformation of a graded Lie group over the spectrum Spec /GL of the ground algebra.     

Kerr metric in the deSitter background

In addition to the Kerr metric with cosmological constant Λ several other metrics are presented giving a Kerr-like solution of Einstein’s equations in the background of deSitter universe. A new metric of what may be termed as rotating deSitter space-time—a space-time devoid of matter but containing null fluid with twisting null rays, has been presented. This metric reduces to the standard deSitter metric when the twist in the rays vanishes. Kerr metric in this background is the immediate generalization of Schwarzschild’s exterior metric with cosmological constant.     

On the pre‐metric foundations of wave mechanics I: massless waves

The mechanics of wave motion in a medium are founded in conservation laws for the physical quantities that the waves carry, combined with the constitutive laws of the medium, and define Lorentzian structures only in degenerate cases of the dispersion laws that follow from the field equations. It is suggested that the transition from wave motion to point motion is best factored into an intermediate step of extended matter motion, which then makes the dimension‐codimension duality of waves and trajectories a natural consequence of the bicharacteristic (geodesic) foliation associated with the dispersion law. This process is illustrated in the conventional case of quadratic dispersion laws, as well as quartic ones, which include the Heisenberg–Euler dispersion law. It is suggested that the contributions to geodesic motion from the non‐quadratic nature of a dispersion law might represent another source of quantum fluctuations about classical extremals, in addition to the diffraction effects that are left out by the geometrical optics approximation.     

度量算符对Gauss编织态的本征作用及自旋几何

对圈量子引力中标架度量矩阵算符对Gauss编织态的作用为本征作用，提供了完整的证明.求得了全部标架度量矩阵算符的表示矩阵，及其期望值.利用自旋几何定理，在内腿颜色k=0和k=2两种情况下，算得了Gauss编织态顶角毗邻的4条腿(P=1)的相位位形切方向间的全部夹角，以及切矢量的长度. 关键词： 度量算符的表示矩阵 度量期望值 切方向间夹角 切矢量长度     

On quantum symmetries of compact metric spaces

An action of a compact quantum group on a compact metric space $(X,d)$ is (D)-isometric if the distance function is preserved by a diagonal action on $X\times X$. In this study, we show that an isometric action in this sense has the following additional property: the corresponding action on the algebra of continuous functions on $X$ by the convolution semigroup of probability measures on the quantum group contracts Lipschitz constants. In other words, it is isometric in another sense due to Li, Quaegebeur, and Sabbe, which partially answers a question posed by Goswami. We also introduce other possible notions of isometric quantum actions in terms of the Wasserstein $p$-distances between probability measures on $X$ for $p\ge 1$, which are used extensively in optimal transportation. Indeed, all of these definitions of quantum isometry belong to a hierarchy of implications, where the two described above lie at the extreme ends of the hierarchy. We conjecture that they are all equivalent.     

Some extensions of the Einstein–Dirac equation

We considered an extension of the standard functional for the Einstein–Dirac equation where the Dirac operator is replaced by the square of the Dirac operator and a real parameter controlling the length of spinors is introduced. For one distinguished value of the parameter, the resulting Euler–Lagrange equations provide a new type of Einstein–Dirac coupling. We establish a special method for constructing global smooth solutions of a newly derived Einstein–Dirac system called the CL-Einstein–Dirac equation of type II (see Definition 3.1).     

Kerr-Newman metric in deSitter background

In addition to the Kerr-Newman metric with cosmological constant several other metrics are presented giving Kerr-Newman type solutions of Einstein-Maxwell field equations in the background of deSitter universe. The electromagnetic field in all the solutions is assumed to be source-free. A new metric of what may be termed as an electrovac rotating de-Sitter space-time—a space-time devoid of matter but containing source-free electromagnetic field and a null fluid with twisting rays—has been presented. In the absence of the electromagnetic field, our solutions reduce to those discussed by Vaidya.     

五维时空中度规的计算

在四维R+R2引力理论中给出了Wheeler-Dewitt(W-D)方程,通过分离变量法得到了W-D方程的解.利用Kaluza-Klein理论将Robertson-Walker度规推广到五维时空,结合时空中的场方程得到宇宙项与能量之间的关系.     

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold admitting parallel one-form

An estimate for the first eigenvalue of the Dirac operator on compact Riemannian spin manifold of positive scalar curvature admitting a parallel one-form is found. The possible universal covering spaces of the manifolds on which the smalles possible eigenvalue is attained are also listed. Moreover, a complete classification of the compact odd-dimensional manifolds whose universal covering space is Sⁿ⁻¹ × is given in the limiting case. All such manifolds are diffeomorphic but not necessarily isometric to Sⁿ⁻¹ × S¹.     

Magnetic fields in 2D and 3D sphere

In this note we study the Landau–Hall problem in the 2D and 3D unit sphere, that is, the motion of a charged particle in the presence of a static magnetic field. The magnetic flow is completely determined for any Riemannian surface of constant Gauss curvature, in particular, the unit 2D sphere. For the 3D case we consider Killing magnetic fields on the unit sphere, and we show that the magnetic flowlines are helices with the given Killing vector field as its axis.     

Invariant metric for nonlinear symplectic maps

In this paper, we construct an invariant metric in the space of homogeneous polynomials of a given degree (≥3). The homogeneous polynomials specify a nonlinear symplectic map which in turn represents a Hamiltonian system. By minimizing the norm constructed out of this metric as a function of system parameters, we demonstrate that the performance of a nonlinear Hamiltonian system is enhanced.     

Accelerated Levi-Civita-Bertotti-Robinson metric in D dimensions

A conformally flat accelerated charge metric is found in an arbitrary dimension D. It is a solution of the Einstein-Maxwell-null fluid equations with a cosmological constant in D ≥ 4 dimensions. When the acceleration is zero, our solution reduces to the Levi-Civita-Bertotti-Robinson metric. We show that the charge loses its energy, for all dimensions, due to the acceleration.     

Black holes: a physical route to the Kerr metric

As a consequence of Birkhoff's theorem, the exterior gravitational field of a spherically symmetric star or black hole is always given by the Schwarzschild metric. In contrast, the exterior gravitational field of a rotating (axisymmetric) star differs, in general, from the Kerr metric, which describes a stationary, rotating black hole. In this paper I discuss the possibility of a quasi–stationary transition from rotating equilibrium configurations of normal matter to rotating bla ck holes.     

Revealing Chern number from quantum metric

Chern number is usually characterized by Berry curvature. Here, by investigating the Dirac model of even-dimensional Chern insulator, we give the general relation between Berry curvature and quantum metric, which indicates that the Chern number can be encoded in quantum metric as well as the surface area of the Brillouin zone on the hypersphere embedded in Euclidean parameter space. We find that there is a corresponding relationship between the quantum metric and the metric on such a hypersphere. We give the geometrical property of quantum metric. Besides, we give a protocol to measure the quantum metric in the degenerate system.