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 $X\in\fr{m}_1, Y\in\fr{m}_2, Z\in\fr{m}_3$ ($o=eK$), 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).

Journal of Geometry (2015)

References in corpus (2)

• Classification of generalized Wallach spaces
• Invariant Einstein metrics on three-locally-symmetric spaces

Cited by in corpus (3)

• Two-step homogeneous geodesics in homogeneous spaces
• Riemannian $M$-spaces with homogeneous geodesics
• Maxima of Curvature Functionals and the Prescribed Ricci Curvature Problem on Homogeneous Spaces