Geodesic orbit metrics on compact simple Lie groups arising from flag manifolds

Abstract

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups

“…by G and right-invariant by K. We also recall that a homogeneous space G/K is called strongly isotropy irreducible if the connected component of K acts irreducibly on the tangent space T eK (G/K) ( [35]). Our second main result is the following characterization, which is partially a consequence of Theorem 1.4 and generalizes the main theorem in [16].…”

“…Finally, we will need the main result in [16]. Before we state this result, consider a compact simple Lie group G and a connected subgroup K of G such that G/K is a generalized flag manifold, i.e.…”

“…where Q is the negative of the Killing form of g and ( , )| z(k)×z(k) is any inner product on z(k). The main result in [16] asserts that any g.o. metric of the form (7.23) is G × K-naturally reductive.…”

“…Given that the metrics (7.23) exhaust the G × K-invariant g.o. metrics on G, the main result in [16] can be restated as follows.…”

“…• Any non-naturally reductive Einstein Lie group induced from one of the standard triples (sp(2n 1 n 2 ), sp(n 1 n 2 )×sp(n 1 n 2 ), 2n 2 sp(n 1 )), (f 4 , so (9), so(8)), (e 8 , so (16), 8su(2)), (e 8 , so (16), 2so(8)) in [38]. Then (G, g) is not a geodesic orbit manifold.…”