Inverse problem of left invariant sprays on Lie groups

Author:

Huang Libing12,Mo Xiaohuan12ORCID

Affiliation:

1. School of Mathematical Sciences, Nankai University, Tianjin 300071, P. R. China

2. Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing 100871, P. R. China

Abstract

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

Funder

National Natural Science Foundation of China

Publisher

World Scientific Pub Co Pte Lt

Subject

General Mathematics

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Parallel Translations for a Left Invariant Spray;Bulletin of the Iranian Mathematical Society;2023-03-09

2. Submersion and Homogeneous Spray Geometry;The Journal of Geometric Analysis;2022-03-26

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