Manifolds of nonpositive curvature and their buildings.

*(English)*Zbl 0643.53037Let \(\tilde M\) denote a complete, simply connected manifold of bounded nonpositive sectional curvature that is the Riemannian covering of a complete manifold M of finite volume. Let \(S\tilde M\) denote the unit tangent bundle of \(\tilde M.\) For each vector \(v\in S\tilde M\) let \(\gamma_ v\) denote the geodesic determined by v and let r(v) denote the dimension of the vector space of all parallel Jacobi vector fields on \(\gamma_ v\). The rank of \(\tilde M,\) denoted rank \((\tilde M)\), is defined to be the minimum value of the integers r(v), \(v\in S\tilde M\). In this paper the authors prove the following result obtained independently with another method by W. Ballmann in Ann. Math., II. Ser. 122, 597-609 (1985; Zbl 0585.53031). Theorem. Let \(\tilde M\) be as above and assume that \(\tilde M\) is irreducible with rank\((\tilde M)=k\geq 2\). Then \(\tilde M\) is a symmetric space of noncompact type and rank k.

The preliminary stages of the proof use or modify results from W. Ballmann, M. Brin and the reviewer [Ann. Math., II. Ser. 122, 171- 203 (1985; Zbl 0589.53047)] and W. Ballmann, M. Brin and R. Spatzier [ibid., 205-235 (1985; Zbl 0598.53046)]. Following Ballmann, Brin and the reviewer [loc. cit.], a vector \(v\in S\tilde M\) is regular if \(r(v)=rank(\tilde M)\). From this work one knows that the set \({\mathcal R}\) of regular unit vectors is dense and open in \(S\tilde M\) and if \(v\in {\mathcal R}\) then the set F(v), the union of all geodesics of \(\tilde M\) parallel to \(\gamma_ v\), is a complete, totally geodesic flat submanifold of \(\tilde M\) of dimension \(k=\text{rank}(\tilde M)\). Every regular vector \(v\in S\tilde M\) determines a Weylsimplex \({\mathcal C}(v)\subseteq S_ vF(v)\) with \(v\in {\mathcal C}(v)\). More generally one may define \({\mathcal C}(v)\) for any vector \(v\in S\tilde M\) that is asymptotic to a regular vector of \(S\tilde M\). If \(\tilde M\) is a symmetric space of noncompact type and rank \(k\geq 2\), then these Weyl simplices are all isometric.

The authors in this paper and Ballmann in loc. cit. prove similar weakened versions of this fact. Ballmann continues by showing that if v is the center of gravity of an appropriate Weyl simplex \({\mathcal C}(w)\), then the orbit of v under the holonomy group at the base point \(\pi(v)\) is a proper closed subset of the unit vectors at \(\pi(v)\). The theorem stated above then follows from a result of M. Berger. The authors follow a different path and define Weyl simplices in the boundary sphere at infinity \(\tilde M(\infty)\) by projecting the Weyl simplices \({\mathcal C}(v)\) onto \(\tilde M(\infty)\). The Weyl simplices in \(\tilde M(\infty)\) form a topological Tits building \(\Delta(\tilde M)\) covering \(\tilde M(\infty)\) and the main result of the authors’ paper “On topological Tits buildings and their classification” [see the review above (Zbl 0643.53036)] shows that \(\Delta(\tilde M)\) is the Tits building associated to a connected, noncompact simple Lie group G. The group G is the connected group of all automorphisms of \(\Delta(\tilde M)\) that are also homeomorphisms of \(\Delta(\tilde M)\). Using arguments similar to those used by Gromov in his extension of the Mostow rigidity theorem the authors then show that \(\tilde M\) is isometric to the symmetric space G/K, where K is a maximal compact subgroup of G.

The preliminary stages of the proof use or modify results from W. Ballmann, M. Brin and the reviewer [Ann. Math., II. Ser. 122, 171- 203 (1985; Zbl 0589.53047)] and W. Ballmann, M. Brin and R. Spatzier [ibid., 205-235 (1985; Zbl 0598.53046)]. Following Ballmann, Brin and the reviewer [loc. cit.], a vector \(v\in S\tilde M\) is regular if \(r(v)=rank(\tilde M)\). From this work one knows that the set \({\mathcal R}\) of regular unit vectors is dense and open in \(S\tilde M\) and if \(v\in {\mathcal R}\) then the set F(v), the union of all geodesics of \(\tilde M\) parallel to \(\gamma_ v\), is a complete, totally geodesic flat submanifold of \(\tilde M\) of dimension \(k=\text{rank}(\tilde M)\). Every regular vector \(v\in S\tilde M\) determines a Weylsimplex \({\mathcal C}(v)\subseteq S_ vF(v)\) with \(v\in {\mathcal C}(v)\). More generally one may define \({\mathcal C}(v)\) for any vector \(v\in S\tilde M\) that is asymptotic to a regular vector of \(S\tilde M\). If \(\tilde M\) is a symmetric space of noncompact type and rank \(k\geq 2\), then these Weyl simplices are all isometric.

The authors in this paper and Ballmann in loc. cit. prove similar weakened versions of this fact. Ballmann continues by showing that if v is the center of gravity of an appropriate Weyl simplex \({\mathcal C}(w)\), then the orbit of v under the holonomy group at the base point \(\pi(v)\) is a proper closed subset of the unit vectors at \(\pi(v)\). The theorem stated above then follows from a result of M. Berger. The authors follow a different path and define Weyl simplices in the boundary sphere at infinity \(\tilde M(\infty)\) by projecting the Weyl simplices \({\mathcal C}(v)\) onto \(\tilde M(\infty)\). The Weyl simplices in \(\tilde M(\infty)\) form a topological Tits building \(\Delta(\tilde M)\) covering \(\tilde M(\infty)\) and the main result of the authors’ paper “On topological Tits buildings and their classification” [see the review above (Zbl 0643.53036)] shows that \(\Delta(\tilde M)\) is the Tits building associated to a connected, noncompact simple Lie group G. The group G is the connected group of all automorphisms of \(\Delta(\tilde M)\) that are also homeomorphisms of \(\Delta(\tilde M)\). Using arguments similar to those used by Gromov in his extension of the Mostow rigidity theorem the authors then show that \(\tilde M\) is isometric to the symmetric space G/K, where K is a maximal compact subgroup of G.

Reviewer: P.Eberlein

##### MSC:

53C35 | Differential geometry of symmetric spaces |

51H20 | Topological geometries on manifolds |

22E46 | Semisimple Lie groups and their representations |

##### Keywords:

nonpositive sectional curvature; finite volume; noncompact type; Tits buildings; symmetric space
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\textit{K. Burns} and \textit{R. Spatzier}, Publ. Math., Inst. Hautes Étud. Sci. 65, 35--59 (1987; Zbl 0643.53037)

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##### References:

[1] | W. Ballmann, Nonpositively curved manifolds of higher rank,Ann. of Math.,122 (1985), 597–609. · Zbl 0585.53031 |

[2] | W. Ballmann, M. Brin andP. Eberlein, Structure of manifolds of non-positive curvature. I,Ann. of Math. 122 (1985), 171–203. · Zbl 0589.53047 |

[3] | W. Ballmann, M. Brin andR. Spatzier, Structure of manifolds of non-positive curvature. II,Ann. of Math.,122 (1985), 205–235. · Zbl 0598.53046 |

[4] | M. Berger, Sur les groupes d’holonomie homogène des variétés à connexion affine et des variétés riemanniennes,Bull. Soc. Math. France,83 (1955), 279–330. · Zbl 0068.36002 |

[5] | W. Ballmann, M. Gromov andV. Schroeder, Manifolds of Nonpositive Curvature, Birkhäuser,Progress in Mathematics,61 (1985). · Zbl 0591.53001 |

[6] | K. Burns andR. Spatzier, On topological Tits buildings and their classification,Publ. Math. I.H.E.S.,65 (1987), 5–34. · Zbl 0643.53036 |

[7] | N. Bourbaki, Groupes et Algèbres de Lie, chap. IV, V, VI,Eléments de Mathématique, fasc. XXXIV, Paris, Hermann (1968). |

[8] | P. Eberlein,Rigidity problems of manifolds of nonpositive curvature, Preprint (1985). · Zbl 0569.53020 |

[9] | P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature. II,Acta Mathematica,149 (1982), 41–69. · Zbl 0511.53048 |

[10] | P. Eberlein andB. O’Neill, Visibility manifolds,Pacific J. Math.,46 (1973), 45–109. · Zbl 0264.53026 |

[11] | S. Kobayashi andK. Nomizu,Foundations of Differential Geometry, vol. I, New York, Wiley (1963). · Zbl 0119.37502 |

[12] | G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces,Annals of Math. Studies, No. 78, Princeton, New Jersey, Princeton University Press (1973). · Zbl 0265.53039 |

[13] | G. A. Margulis, Discrete Groups of Motions of Manifolds of Nonpositive Curvature,A.M.S. Translations,109 (1977), 33–45. · Zbl 0367.57012 |

[14] | R. J. Spatzier,The geodesic flow and an approach to the classification of manifolds of nonpositive curvature, M.S.R.I. Berkeley Preprint 004-84 (1984). |

[15] | J. Simons, On transitivity of holonomy systems,Ann. of Math.,76 (1962), 213–234. · Zbl 0106.15201 |

[16] | E. H. Spanier,Algebraic Topology, McGraw-Hill (1966). |

[17] | J. Tits, Buildings of Spherical Type and Finite BN-pairs,Springer Lecture Notes in Mathematics,386 (1970). · Zbl 0295.20047 |

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