Flows with periodic factors on homogeneous spaces (Q1084730)

We prove that on any homogeneous space G/H, where G is a connected Lie group of dimension \(\geq 4\) and H is a closed subgroup of G, there exists a flow, induced by a nontrivial one-parameter subgroup \((g_ t)\) of G, which admits nontransitive periodic factors; that is, for some closed subgroup K containing H the action of \((g_ t)\) on G/K is periodic but not transitive. In particular, such a flow does not admit any dense orbits. By analyzing the possibilities in lower dimensions we also deduce that the latter conclusion can be upheld for a suitable nontrivial one- parameter subgroup whenever G/H is of dimension at least 2. The question arose out of a result of \textit{C. Scheiderer} [cf. Monatsh. Math. 98, 75- 81 (1984; Zbl 0543.22003)] asserting existence of a \((g_ t)\) for which not all orbits are dense, provided G/H is of dimension at least 2.