{
  "id": "1904.07288",
  "version": "v1",
  "published": "2019-04-15T18:54:38.000Z",
  "updated": "2019-04-15T18:54:38.000Z",
  "title": "Minimal codimension one foliation of a symmetric space by Damek-Ricci spaces",
  "authors": [
    "Gerhard Knieper",
    "John R. Parker",
    "Norbert Peyerimhoff"
  ],
  "comment": "16 pages",
  "categories": [
    "math.DG"
  ],
  "abstract": "In this article we consider solvable hypersurfaces of the form $N \\exp(\\R H)$ with induced metrics in the symmetric space $M = SL(3,\\C)/SU(3)$, where $H$ a suitable unit length vector in the subgroup $A$ of the Iwasawa decomposition $SL(3,\\C) = NAK$. Since $M$ is rank $2$, $A$ is $2$-dimensional and we can parametrize these hypersurfaces via an angle $\\alpha \\in [0,\\pi/2]$ determining the direction of $H$. We show that one of the hypersurfaces (corresponding to $\\alpha = 0$) is minimally embedded and isometric to the non-symmetric $7$-dimensional Damek-Ricci space. We also provide an explicit formula for the Ricci curvature of these hypersurfaces and show that all hypersurfaces for $\\alpha \\in (0,\\frac{\\pi}{2}]$ admit planes of both negative and positive sectional curvature. Moreover, the symmetric space $M$ admits a minimal foliation with all leaves isometric to the non-symmetric $7$-dimensional Damek-Ricci space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-15T18:54:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric space",
      "minimal codimension",
      "dimensional damek-ricci space",
      "hypersurfaces",
      "suitable unit length vector"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}