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.
Huihui An, Zaili Yan, Shaoxiang Zhang
Liaoning Normal University; Ningbo University; Shandong University of Science and Technology
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).
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