关于广义 Douglas-Weyl 喷射的一个注记

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 43-52.

关于广义 Douglas-Weyl 喷射的一个注记

郑大小()

安徽师范大学数学与统计学院安徽芜湖 241002

收稿日期:2022-03-07 修回日期:2022-07-22 出版日期:2023-02-26 发布日期:2023-03-07
作者简介:郑大小, E-mail: 2015046@ahnu.edu.cn
基金资助:
安徽省自然科学青年基金(2008085QA05)

A Note on Generalized Douglas-Weyl Spray

Zheng Daxiao()

Department of Mathematics and Statistics Science, Anhui Normal University, Anhui Wuhu 241002

Received:2022-03-07 Revised:2022-07-22 Online:2023-02-26 Published:2023-03-07
Supported by:
The AHNSF(2008085QA05)

摘要/Abstract

摘要：

该文研究广义 Douglas-Weyl 喷射. 证明了一个喷射 $G$ 是广义 Douglas-Weyl 喷射当且仅当它的 Weyl 张量是二次型的. 由此得到一个推论, 一个芬斯勒度量是广义Douglas-Weyl 度量当且仅当它的 Weyl 张量是二次型的. 进一步, 该文研究具有二次型的黎曼曲率张量的喷射, 证明了一个喷射具有二次型的黎曼曲率张量当且仅当 $\dot{B}^{~i}_{j~kl}=0$.

关键词: 喷射, Weyl张量, Douglas张量

Abstract:

In this paper, we study Generalized Douglas-Weyl spray. We show that a spray $G$ is a Generalized Douglas-Weyl spray if and only if it is $W$-quadratic. As a corollary, we show that a Finsler metric $F$ is a Generalized Douglas-Weyl metric if and only if it is $W$-quadratic. Furthermore, we consider $R$-quadratic spray and prove that a spray $G$ is $R$-quadratic if and only if $\dot{B}^{~i}_{j~kl}=0$.

Key words: Spray, Weyl curvature, Douglas Tensor

中图分类号:

O184

郑大小. 关于广义 Douglas-Weyl 喷射的一个注记[J]. 数学物理学报, 2023, 43(1): 43-52.

Zheng Daxiao. A Note on Generalized Douglas-Weyl Spray[J]. Acta mathematica scientia,Series A, 2023, 43(1): 43-52.

参考文献 13

[1]	Bácsó S, Papp I. A note on a generalized Douglas space. Period Math Hungar, 2004, 48: 181-184 doi: 10.1023/B:MAHU.0000038974.24588.83
[2]	Douglas J. The general geometry of paths. Ann Math, 1927, 29: 143-168 doi: 10.2307/1967989
[3]	Li B, Shen Z. Randers metrics of quadratic Riemann curvature. Int J Math, 2009, 20: 369-376 doi: 10.1142/S0129167X09005315
[4]	Li B, Shen Z. Ricci curvature tensor and non-Riemannian quantities. Can Math Bull, 2015, 58: 530-537 doi: 10.4153/CMB-2014-063-4
[5]	Li B, Shen Z. Sprays of isotropic curvature. Int J Math, 2018, 29: 1850003 doi: 10.1142/S0129167X18500039
[6]	Li Y, Mo X, Yu Y. Inverse problem of sprays with scalar curvature. Int J Math, 2019, 30: 1950041 doi: 10.1142/S0129167X19500411
[7]	Li Y, Mo X. On Sakaguchi-type result in projective spray geometry. Ann Mat Pura Appl, 2021, 200: 1-9 doi: 10.1007/s10231-020-00980-9
[8]	Mo X, Zhu H. On a projective class of Finsler metrics with orthogonal invariance. Difer Geom Appl, 2017, 52: 167-180
[9]	Najafi B, Shen Z, Tayebi A. Finsler metrics of scalar flag curvature with special non-Riemannian curvature properties. Geom Dedicata, 2008, 131: 87-97 doi: 10.1007/s10711-007-9218-9
[10]	Sakaguchi T. On Finsler spaces of scalar curvature. Progr Math, 1982, 38: 75-85 doi: 10.1007/BF02591853
[11]	Shen Z. Diferential Geometry of Spray and Finsler Spaces. Dordrecht: Kluwer Academic Publishers, 2001
[12]	Tatebi A, Sadeghi H. On generalized Douglas-Weyl$(\alpha, \beta)$-metrics. Acta Math Sinica, 2015, 10: 1611-1620
[13]	Yang G. Some classes of sprays in projective spray geometry. Difer Geom Appl, 2011, 29: 606-614