An introduction to the theory of generalized conics and their applications

JOURNAL OF GEOMETRY AND PHYSICS

卷:61

Article

Vincze, Cs.¹  Nagy, A.¹ 

[1] Univ Debrecen, Inst Math, H-4010 Debrecen, Hungary

关键词: Minkowski functionals;    Finsler manifolds;    Berwald manifolds;   

DOI  :  10.1016/j.geomphys.2010.12.003

来源: Elsevier

【 摘 要 】

A generalized conic is a set of points with the same average distance from the pointset Gamma in the Euclidean coordinate space. The measuring of the average distance is realized via integration over Gamma as the set of foci. Using generalized conics we give a process for constructing convex bodies which are invariant under a fixed subgroup G of the orthogonal group in R(n). The motivation is to present the existence of non-Euclidean Minkowski functionals with G subset of O(n) in the linear isometry group provided that the closure of G is not transitive on the unit sphere. As an application, consider R(n) as the tangent space at a point of a connected Riemannian manifold M and G as the holonomy group. If the holonomy group is not transitive on the unit sphere in the tangent space, then the Levi-Civita connection is (re)metrizable in the sense that there is a smooth collection of non-Euclidean Minkowski functionals on the tangent spaces such that it is invariant under parallel transport with respect to the Levi-Civita connection (according to Berger's list of possible Riemannian holonomy groups, all of them are transitive on the unit sphere in the tangent space except in the case where the manifold is a symmetric space of rank >= 2). We present the (re)metrizability theorem in a more general context of metrical linear connections with a torsion tensor that is not necessarily vanishing. This allows us to declare eight classes of manifolds equipped with an invariant smooth collection of Minkowski functionals on the tangent spaces. They are called Berwald manifolds in a general sense. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free