Bulletin of the
Korean Mathematical Society BKMS

In this paper, we mainly study the problem of the existence of homogeneous geodesics in sub-Finsler manifolds. Firstly, we obtain a characterization of a homogeneous curve to be a geodesic. Then we show that every compact connected homogeneous sub-Finsler manifold and Carnot group admits at least one homogeneous geodesic through each point. Finally, we study a special class of $\ell^{p}$-type bi-invariant metrics on compact semi-simple Lie groups. We show that every homogeneous curve in such a metric space is a geodesic. Moreover, we prove that the Alexandrov curvature of the metric space is neither non-positive nor non-negative.

Keywords: Sub-Finsler manifolds, homogeneous geodesics, bi-invariant metrics

MSC numbers: Primary 53C17, 53C60, 53C22

Supported by: This work was financially supported by the Fundamental Research Funds for the Provincial Universities of Zhejiang.