Riemannian $M$-spaces with homogeneous geodesics

TL;DR

This paper studies homogeneous geodesics on $M$-spaces derived from flag manifolds, identifying conditions under which the standard metric is the unique geodesic orbit metric and characterizing other g.o. metrics.

Contribution

It classifies g.o. metrics on $M$-spaces associated with flag manifolds, providing necessary and sufficient conditions for their existence beyond the standard metric.

Findings

Standard metric is the only g.o. metric in certain classes of $M$-spaces.

Necessary and sufficient conditions for g.o. metrics are established for other classes.

Analysis based on isotropy representation decompositions of the tangent space.

Abstract

We investigate homogeneous geodesics in a class of homogeneous spaces called $M$-spaces, which are defined as follows. Let $G/K$ be a generalized flag manifold with $K=C(S)=S\times K_1$, where $S$ is a torus in a compact simple Lie group $G$ and $K_1$ is the semisimple part of $K$. Then the {\it associated $M$-space} is the homogeneous space $G/K_1$. These spaces were introduced and studied by H.C. Wang in 1954. We prove that for various classes of $M$-spaces the only g.o. metric is the standard metric. For other classes of $M$-spaces we give either necessary, or necessary and sufficient conditions, so that a $G$-invariant metric on $G/K_1$ is a g.o. metric. The analysis is based on properties of the isotropy representation $\mathfrak{m}=\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the flag manifold $G/K$ (as Ad$(K)$-modules) and corresponding decomposition $\mathfrak{n}=\mathfrak{s}\oplus\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the tangent space of the $M$-space $G/K_1$ (as Ad$(K_1)$-modules).