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 $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\frak{m}_i,\frak{m}_i]\subset \frak{k}$. Assuming that the submodules $\frak{m}_i$ are pairwise non isomorphic, we study geodesics on such spaces of the form $\gamma (t)=\exp (tX)\exp (tY)\exp (tZ)\cdot o$, where $X\in\fr{m}_1, Y\in\fr{m}_2, Z\in\fr{m}_3$ ($o=eK$), 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).