Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2023; 60(1): 33-46

Online first article January 25, 2023      Printed January 31, 2023

https://doi.org/10.4134/BKMS.b210835

Copyright © The Korean Mathematical Society.

S-curvature and geodesic orbit property of invariant $(\alpha_{1},\alpha_{2})$-metrics on spheres

Huihui An, Zaili Yan, Shaoxiang Zhang

Liaoning Normal University; Ningbo University; Shandong University of Science and Technology

Abstract

Geodesic orbit spaces are homogeneous Finsler spaces whose geodesics are all orbits of one-parameter subgroups of isometries. Such Finsler spaces have vanishing S-curvature and hold the Bishop-Gromov volume comparison theorem. In this paper, we obtain a complete description of invariant $(\alpha_{1},\alpha_{2})$-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit $(\alpha_{1},\alpha_{2})$-metrics on spheres. We mainly show that a ${\mathrm S}{\mathrm p}(n+1)$-invariant $(\alpha_{1},\alpha_{2})$-metric on $\mathrm{S}^{4n+3}={\mathrm S}{\mathrm p}(n+1)/{\mathrm S}{\mathrm p}(n)$ is geodesic orbit with respect to ${\mathrm S}{\mathrm p}(n+1)$ if and only if it is ${\mathrm S}{\mathrm p}(n+1){\mathrm S}{\mathrm p}(1)$-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.

Keywords: Finsler geodesic orbit space, $(\alpha_{1},\alpha_{2})$-metric, S-curvature

MSC numbers: Primary 53C60, 53C30, 53C25

Supported by: This work was financially supported by the Fundamental Research Funds for the Provincial Universities of Zhejiang, National Natural Science Foundation of China (No. 12201358), Natural Science Foundation of Shandong Province (No. ZR2021QA051).

Stats or Metrics

Share this article on :

Related articles in BKMS

more +