Geodesics in generalized Wallach spaces

TL;DR

This paper investigates geodesics in generalized Wallach spaces, focusing on their expression as orbits of exponential maps, and applies the findings to specific homogeneous spaces like flag manifolds and Stiefel manifolds.

Contribution

It provides a new perspective on geodesics in generalized Wallach spaces using exponential orbit representations and Riemannian submersions, with applications to flag and Stiefel manifolds.

Findings

Characterization of geodesics as orbits of exponential maps under certain metrics

Description of geodesics in specific homogeneous spaces like flag and Stiefel manifolds

Relation of results to geodesic orbit spaces (g.o. 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).