Homogeneous geodesics in the flag manifold [formula omitted

Abstract

A geodesic curve in a Riemannian homogeneous manifold ( M = G / K , g ) is called a homogeneous geodesic if it is an orbit of an one-parameter subgroup of the Lie group G . We investigate G -invariant metrics such that all geodesics are homogeneous for the flag manifold M = SO ( 2 l + 1 ) / U ( l - m ) × SO ( 2 m + 1 ) . By reformulating the problem into a matrix form we show that SO ( 2 ℓ + 1 ) / U ( ℓ - m ) × SO ( 2 m + 1 ) has homogeneous geodesics with respect to any SO ( 2 ℓ + 1 ) -invariant metric if and only if m = 0 . In all other cases this space admits at least one non-homogeneous geodesic. We also give examples of finding homogeneous geodesics in the above flag manifold for special values of l and m .