{
  "id": "1205.6222",
  "version": "v2",
  "published": "2012-05-28T22:07:28.000Z",
  "updated": "2015-02-26T01:48:13.000Z",
  "title": "Tits Geometry and Positive Curvature",
  "authors": [
    "Fuquan Fang",
    "Karsten Grove",
    "Gudlaugur Thorbergsson"
  ],
  "comment": "63 pages",
  "categories": [
    "math.DG",
    "math.GR",
    "math.GT"
  ],
  "abstract": "There is a well known link between (maximal) polar representations and isotropy representations of symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns-Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and irreducible topological spherical buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected closed positively curved manifold. Although this chamber system is typically not a Tits geometry of type M, we prove that in all cases but two that its universal Tits cover indeed is a building. We construct a topology on this universal cover making it into a compact spherical building in the sense of Burns and Spatzier. Using this structure we classify up to equivariant diffeomorphism all polar actions on (simply connected) positively curved manifolds of cohomogeneity at least two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-28T22:07:28.000Z",
      "abstract": "There is a well known link between (maximal) irreducible polar representations and isotropy representations of irreducible symmetric spaces provided by Dadok. Moreover, the theory by Tits and Burns - Spatzier provides a link between irreducible symmetric spaces of non-compact type of rank at least three and compact topological spherical irreducible buildings of rank at least three. We discover and exploit a rich structure of a (connected) chamber system of finite (Coxeter) type M associated with any polar action of cohomogeneity at least two on any simply connected (closed) positively curved manifold. Although this chamber system is typically not a (Tits) geometry of type M, we prove that in all cases but one that its universal (Tits) cover indeed is a building. We construct a topology on this universal cover making it into a compact topological building in the sense of Burns and Spatzier. Our work shows that the exception indeed provides a new example (also discovered by Lytchak) of a C3 geometry whose universal cover is not a building. We use this structure to prove the following rigidity theorem: Any polar action of cohomogeneity at least two on a simply connected positively curved manifold is smoothly equivalent to a polar action on a rank one symmetric space. The analysis and methods used in the reducible case (including the case of fixed points), the case of cohomogeneity two, and the general irreducible case in cohomogeneity at least three are quite different from one another. Throughout the local approach to buildings by Tits plays a significant role. The present work and different independent work of Lytchak on foliations on symmetric spaces are the first instances where this approach has been used in differential geometry.",
      "comment": "55 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-26T01:48:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tits geometry",
      "positive curvature",
      "polar action",
      "irreducible symmetric spaces",
      "topological spherical irreducible buildings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.6222F"
    }
  }
}