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The symmetric space, strong isotropy irreducibility and equigeodesic properties

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Abstract

A smooth curve on a homogeneous manifold G/H is called a Riemannian equigeodesic if it is a homogeneous geodesic for any G-invariant Riemannian metric. The homogeneous manifold G/H is called Riemannian equigeodesic, if for any x ∈ G/H and any nonzero y ∈ Tx(G/H), there exists a Riemannian equigeodesic c(t) with c(0) = x and ċ(0) = y. These two notions can be naturally transferred to the Finsler setting, which provides the definitions for Finsler equigeodesics and Finsler equigeodesic spaces. We prove two classification theorems for Riemannian equigeodesic spaces and Finsler equigeodesic spaces, respectively. Firstly, a homogeneous manifold G/H with a connected simply connected quasi compact G and a connected H is Riemannian equigeodesic if and only if it can be decomposed as a product of Euclidean factors and compact strongly isotropy irreducible factors. Secondly, a homogeneous manifold G/H with a compact semisimple G is Finsler equigeodesic if and only if it can be locally decomposed as a product, in which each factor is Spin(7)/G2, G2/SU(3) or a symmetric space of compact type. These results imply that the symmetric space and the strongly isotropy irreducible space of compact type can be interpreted by equigeodesic properties. As an application, we classify the homogeneous manifold G/H with a compact semisimple G such that all the G-invariant Finsler metrics on G/H are Berwald. It suggests a new project in homogeneous Finsler geometry, i.e., to systematically study the homogeneous manifold G/H on which all the G-invariant Finsler metrics satisfy a certain geometric property.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12131012, 12001007 and 11821101), Beijing Natural Science Foundation (Grant No. 1222003) and Natural Science Foundation of Anhui Province (Grant No. 1908085QA03). The authors sincerely thank Valerii Berestovskii, Huibin Chen, Zhiqi Chen, Shaoqiang Deng, Yurii Nikonorov, Zaili Yan, Fuhai Zhu and Wolfgang Ziller for helpful discussion.

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Authors and Affiliations

  1. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Ming Xu

  2. School of Microelectronics and Data Science, Anhui University of Technology, Ma’anshan, 243032, China

    Ju Tan

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  1. Ming Xu
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  2. Ju Tan
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Correspondence to Ju Tan.

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Xu, M., Tan, J. The symmetric space, strong isotropy irreducibility and equigeodesic properties. Sci. China Math. 67, 129–148 (2024). https://doi.org/10.1007/s11425-022-2090-1

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