Geodesics in generalized Wallach spaces

Abstract

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces M = G/K whose isotropy representation decomposes into a direct sum of three submodules ${\mathfrak{m}=\mathfrak{m}_1\oplus\mathfrak{m}_2\oplus\mathfrak{m}_3}$ , satisfying the relations ${[\mathfrak{m}_i,\mathfrak{m}_i]\subset \mathfrak{k}}$ . Assuming that the submodules ${\mathfrak{m}_i}$ are pairwise non isomorphic, we study geodesics on such spaces of the form ${\gamma (t)=\exp (tX)\exp (tY)\exp (tZ)\cdot o}$ , where ${X\in\mathfrak{m}_1, Y\in\mathfrak{m}_2, Z\in\mathfrak{m}_3}$ (o = eK), with respect to a G-invariant metric. Our investigation imposes certain restrictions on the G-invariant metric, so the geodesics turn out to be orbits of two exponential terms. We give a point of view using Riemannian submersions. As an application, we describe geodesics in generalized flag manifolds with three isotropy summands and with second Betti number b ₂(M) = 2, and in the Stiefel manifolds SO(n + 2)/S(n). We relate our results to geodesic orbit spaces (g.o. spaces).