{
  "id": "1610.01278",
  "version": "v1",
  "published": "2016-10-05T06:00:09.000Z",
  "updated": "2016-10-05T06:00:09.000Z",
  "title": "Riemannian $M$-spaces with homogeneous geodesics",
  "authors": [
    "Andreas Arvanitoyeorgos",
    "Yu Wang",
    "Guosong Zhao"
  ],
  "comment": "15 pages",
  "categories": [
    "math.DG"
  ],
  "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 $G/K$, in particular on the dimension of the submodules $\\mathfrak{m}_i$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-05T06:00:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C25",
      "53C30"
    ],
    "keywords": [
      "homogeneous geodesics",
      "riemannian",
      "compact simple lie group",
      "homogeneous space",
      "generalized flag manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}