Riemannian $M$-spaces with homogeneous geodesics

Andreas Arvanitoyeorgos, Yu Wang, Guosong Zhao

Published 2016-10-05Version 1

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 $G/K$, in particular on the dimension of the submodules $\mathfrak{m}_i$.