A g.o. Finsler manifold with K≥0 and (FP) is compact, satisfies rk g ≤ rk h +1 when G/H with compact Lie algebra, and even-dimensional cases are coset spaces with positive curvature metrics.
On homogeneous geodesics and weakly symmetric spaces
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abstract
In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its geodesics are homogeneous. We also study general properties of homogeneous geodesics, in particular, the structure of the closure of a given homogeneous geodesic. We present several examples where this closure is a torus of dimension $\geq 2$ which is (respectively, is not) totally geodesic in the ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups supplied with left-invariant Riemannian metrics.
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math.DG 1years
2019 1verdicts
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Geodesic orbit Finsler space with $K\geq0$ and the (FP) condition
A g.o. Finsler manifold with K≥0 and (FP) is compact, satisfies rk g ≤ rk h +1 when G/H with compact Lie algebra, and even-dimensional cases are coset spaces with positive curvature metrics.