Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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S-CURVATURE AND GEODESIC ORBIT PROPERTY OF INVARIANT (α1, α2)-METRICS ON SPHERES

  • Huihui, An (School of Mathematics Liaoning Normal University) ;
  • Zaili, Yan (School of Mathematics and Statistics Ningbo University) ;
  • Shaoxiang, Zhang (College of Mathematics and Systems Science Shandong University of Science and Technology)
  • Received : 2021.11.17
  • Accepted : 2022.10.11
  • Published : 2023.01.31
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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 (α1, α2)-metrics on spheres with vanishing S-curvature. Also, we give a description of invariant geodesic orbit (α1, α2)-metrics on spheres. We mainly show that a Sp(n + 1)-invariant (α1, α2)-metric on S4n+3 = Sp(n + 1)/Sp(n) is geodesic orbit with respect to Sp(n + 1) if and only if it is Sp(n + 1)Sp(1)-invariant. As an interesting consequence, we find infinitely many Finsler spheres with vanishing S-curvature which are not geodesic orbit spaces.

Keywords

Acknowledgement

We are deeply grateful to the reviewers of this paper for very careful reading and useful suggestions.

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