Riemannian $M$-spaces with homogeneous geodesics

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).

28 pages

References in corpus (3)

• Homogeneous geodesics in homogeneous Finsler spaces
• On the structure of geodesic orbit Riemannian spaces
• Two-step homogeneous geodesics in homogeneous spaces