Two-step Homogeneous Geodesics in Homogeneous Spaces

Abstract

We study geodesics of the form γ(t) = π(exp(tX) exp(tY )), X, Y ∈ g = Lie(G), in homogeneous spaces G/K, where π : G → G/K is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of G (i.e. γ(t) = π(exp(tX)), X ∈ g). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for X, Y ∈ m = T o (G/K). We use these conditions to obtain examples of Riemannian homogeneous spaces G/K so that all geodesics of G/K are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and k-symmetric spaces with k-even, equipped with certain one-parameter families of invariant metrics.Mathematics Subject Classification. Primary 53C25; Secondary 53C30.

Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems

“…In the work [9] N.P. Souris and the author considered a generalisation of homogeneous geodesics, namely geodesics of the form…”

“…Geodesics of the form (7) had appeared in the work [65] of H.C. Wang as geodesics in a semisimple Lie group G, equipped with a metric induced by a Cartan involution of the Lie algebra g of G. Also, in [25] R. Dohira proved that if the tangent space T o (G/K) of a homogeneous space splits into submodules m 1 , m 2 satisfying certain algebraic relations, and if G/K is endowed with a special one parameter family of Riemannian metrics g c , then all geodesics of the Riemannian space (G/K, g c ) are of the form (7). The main result of [9] is the following: Theorem 9.1. ( [9]) Let M = G/K be a homogeneous space admitting a naturally reductive Riemannian metric.…”

“…The main result of [9] is the following: Theorem 9.1. ( [9]) Let M = G/K be a homogeneous space admitting a naturally reductive Riemannian metric. Let B be the corresponding inner product on m = T o (G/K).…”

“…In [9] we applied the above recipe to the following classes of homogeneous spaces: 1) Lie groups with bi-invariant metrics, equipped with an one parameter family of left-invariant metrics.…”

Riemannian generalized C-spaces with homogeneous geodesics

“…Finally, the notion of homogeneous geodesics can be extended to geodesics which are orbits of a product of two exponential factors (cf. [4], [5]).…”