摘要 :
In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". An indefinite metric polyhedron is a locally finite simplicial complex where each simplex is e...
展开
In this paper, isometric embedding results of Greene, Gromov and Rokhlin are extended to what are called "indefinite metric polyhedra". An indefinite metric polyhedron is a locally finite simplicial complex where each simplex is endowed with a quadratic form (which, in general, is not necessarily positive-definite, or even non-degenerate). It is shown that every indefinite metric polyhedron (with the maximal degree of every vertex bounded above) admits a simplicial isometric embedding into Minkowski space of an appropriate signature. A simple example is given to show that the dimension bounds in the compact case are sharp, and that the assumption on the upper bound of the degrees of vertices cannot be removed. These conditions can be removed though if one allows for isometric embeddings which are merely piecewise linear instead of simplicial.
收起
摘要 :
Back in 1985, Wang and Ziller obtained a complete classification of all homogeneous spaces of compact simple Lie groups on which the standard or Killing metric is Einstein. The list consists, beyond isotropy irreducible spaces, of...
展开
Back in 1985, Wang and Ziller obtained a complete classification of all homogeneous spaces of compact simple Lie groups on which the standard or Killing metric is Einstein. The list consists, beyond isotropy irreducible spaces, of 12 infinite families (two of them are actually conceptual constructions) and 22 isolated examples. We study in this paper the nature of each of these Einstein metrics as a critical point of the scalar curvature functional.
收起
摘要 :
We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 - d) < 1, where d is the Minkowski dimension of the extremal set of K, and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area....
展开
We show that among all plane Hilbert geometries, the hyperbolic plane has maximal volume entropy. More precisely, we show that the volume entropy is bounded above by 2/(3 - d) < 1, where d is the Minkowski dimension of the extremal set of K, and we construct an explicit example of a plane Hilbert geometry with noninteger volume entropy. In arbitrary dimension, the hyperbolic space has maximal entropy among all Hilbert geometries satisfying some additional technical hypothesis. To achieve this result, we construct a new projective invariant of convex bodies, similar to the centroaffine area.
收起
摘要 :
The point pair function p(G) defined in a domain G not subset of R-n is shown to be a quasimetric, and its other properties are studied. For a convex domain G not subset of R-n, a new intrinsic quasi-metric called the function wG ...
展开
The point pair function p(G) defined in a domain G not subset of R-n is shown to be a quasimetric, and its other properties are studied. For a convex domain G not subset of R-n, a new intrinsic quasi-metric called the function wG is introduced. Several sharp results are established for these two quasi-metrics, and their connection to the triangular ratio metric is studied.
收起
摘要 :AbstractA pseudo-Finsleroid metric functionFof the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signatu...
展开AbstractA pseudo-Finsleroid metric functionFof the two-axes structure that involves the vertical axis and the horizontal axis is proposed assuming constancy of the curvature of indicatrix. The curvature is negative and the signature of the Finslerian metric tensor is exactly$$(+-\cdots )$$. The functionFendows the tangent space with the geometry which possesses many interesting Finslerian properties. The use of the angle representation is the underlying method which has been conveniently and successfully applied. The appearance of the positive-definite Finsleroid metric function in the horizontal sections of the tangent space is established.]]>
收起
摘要 :
We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonia...
展开
We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Henon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Henon-Heiles model are obtained. And the numerical results for the Henon-Heiles model show us the instability of the associated geodesic spreads.
收起
摘要 :
This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov-Hausdorff theory based on the use of the Lorentz distance as basic kinematical variable. We ...
展开
This paper is part of a research programme on the structure of the moduli space of Lorentzian geometries, a Lorentzian analogue of Gromov-Hausdorff theory based on the use of the Lorentz distance as basic kinematical variable. We first prove results aimed at a better understanding of the tools available in this framework, such as the relationship between notions of closeness used to define limit spaces, and the properties of the auxiliary 'strong' Riemannian metric defined on each Lorentz space. Then we examine concepts motivated by applications to quantum gravity, namely causality of the limit spaces and compactness of classes of Lorentz spaces.
收起
摘要 :
We study the dependence of three-axes positive-definite Finsleroid metric functions on the Finsleroid azimuthal angle theta in the three-dimensional case, provided the condition of the angle-separation in the involved characterist...
展开
We study the dependence of three-axes positive-definite Finsleroid metric functions on the Finsleroid azimuthal angle theta in the three-dimensional case, provided the condition of the angle-separation in the involved characteristic functions is fulfilled. The complete set of algebraic and differential equations characterizing the class of three-axes positive-definite Finsleroid metric functions is derived and explicit dependence of the involved characteristic functions on the angle theta is obtained.
收起
摘要 :
An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to ...
展开
An extension of conformal equivalence for Finsler metrics is introduced and called weakly conformal equivalence and is used to define the weakly conformal transformations. The conformal Lichnerowicz-Obata conjecture is refined to weakly conformal Finsler geometry. It is proved that: If X is a weakly conformal complete vector field on a connected Finsler space (M, F) of dimension n ≥ 2, then, at least one of the following statements holds: (a) There exists a Finsler metric F_1 weakly conformally equivalent to F such that X is a Killing vector field of the Finsler metric, (b) M is diffeomorphic to the sphere S~n and the Finsler metric is weakly conformally equivalent to the standard Riemannian metric on S~n, and (c) M is diffeomorphic to the Euclidean space R~n and the Finsler metric F is weakly conformally equivalent to a Minkowski metric on R~n. The considerations invite further dynamics on Finsler manifolds.
收起
摘要 :
Given a time function tau on a spacetime M, we define a null distance function, <(d)(tau)over cap>, built from and closely related to the causal structure of M. In basic models with timelike. tau we show that (1) (d(tau)) over cap is a definite dis...
展开
Given a time function tau on a spacetime M, we define a null distance function, <(d)(tau)over cap>, built from and closely related to the causal structure of M. In basic models with timelike. tau we show that (1) (d(tau)) over cap is a definite distance function, which induces the manifold topology, (2) the causal structure of M is completely encoded in (d(tau)) over cap and tau. In general, (d(tau)) over cap is a conformally invariant pseudometric, which may be indefinite. We give an 'anti-Lipschitz' condition on tau, which ensures that (d(tau)) over cap is definite, and show this condition to be satisfied whenever tau has gradient vectors del tau almost everywhere, with del tau locally 'bounded away from the light cones'. As a consequence, we show that the cosmological time function of Andersson et al (1998 Class. Quantum Grav. 15 309-22) is anti-Lipschitz when 'regular', and hence induces a definite null distance function. This provides what may be interpreted as a canonical metric space structure on spacetimes which emanate from a common initial singularity, e.g. a 'big bang'.
收起
原文获取