Symmetry diffeomorphism group of a manifold of nonpositive curvature
Author:
Patrick Eberlein
Journal:
Trans. Amer. Math. Soc. 309 (1988), 355374
MSC:
Primary 53C20
MathSciNet review:
957076
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Abstract: Let denote a complete simply connected manifold of nonpositive sectional curvature. For each point let denote the diffeomorphism of that fixes and reverses all geodesics through . The symmetry diffeomorphism group generated by all diffeomorphisms extends naturally to group of homeomorphisms of the boundary sphere . A subset of is called involutive if it is invariant under . Theorem. Let be a proper, closed involutive subset. For each point let denote the linear span in of those vectors at that are tangent to a geodesic whose asymptotic equivalence class belongs to . If is a proper subspace of for some point , then splits as a Riemannian product such that is the distribution of induced by . This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if is irreducible and admits a proper, closed involutive subset , then is isometric to a symmetric space of noncompact type and rank .
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 [BBE]
 W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature, I, Ann. of Math. (2) 122 (1985), 171203. MR 799256 (87c:58092a)
 [BGS]
 W. Ballmann, M. Gromov and V. Schroeder, Manifolds of nonpositive curvature, Birkhäuser, 1985. MR 823981 (87h:53050)
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 R. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 149. MR 0251664 (40:4891)
 [BS]
 K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Publ. Inst. Hautes Etudes Sci. 65 (1987), 3559. MR 908215 (88g:53050)
 [CE]
 S. Chen and P. Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980), 73103. MR 550653 (82k:53052)
 [E1]
 P. Eberlein, Euclidean de Rham factor of a lattice of nonpositive curvature, J. Differential Geometry 18 (1983), 209220. MR 710052 (84i:53042)
 [E2]
 , Geodesic flows on negatively curved manifolds, I, Ann. of Math. (2) 95 (1972), 492510. MR 0310926 (46:10024)
 [E3]
 , Isometry groups of simply connected manifolds of nonpositive curvature, II, Acta Math. 149 (1982), 4169. MR 674166 (83m:53055)
 [E4]
 , subgroups in spaces of nonpositive curvature, Curvature and Topology of Riemannian Manifolds, Proc. Katata, 1985, Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 4188. MR 859576 (87k:53088)
 [E5]
 , Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), 435476. MR 577132 (82m:53040)
 [E6]
 , Rigidity of lattices of nonpositive curvature, Ergodic Theory Dynamical Systems 3 (1983), 4785. MR 743028 (86f:53049)
 [E7]
 , Structure of manifolds of nonpositive curvature, Global Differential Geometry and Global Analysis, 1984, Lecture Notes in Math., vol. 1156, Springer, Berlin, 1985, pp. 86153. MR 824064 (87d:53080)
 [E8]
 , Symmetry diffeomorphism group of a manifold of nonpositive curvature, II (submitted).
 [EO]
 P. Eberlein and B. O'Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45109. MR 0336648 (49:1421)
 [GW]
 D. Gromoll and J. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545552. MR 0281122 (43:6841)
 [H]
 S. Helgason, Differential geometry and symmetric spaces, Academic Press, New York, 1962. MR 0145455 (26:2986)
 [K]
 F. I. Karpelevic, The geometry of geodesics and the eigenfunctions of the BeltramiLaplace operator on symmetric spaces, Trans. Moscow Math. Soc. 14 (1965), 51199. MR 0231321 (37:6876)
 [KN]
 S. Kobayashi and K. Nomizu, Foundations of differential geometry, vol. 1, Wiley, New York, 1963, pp. 179193.
 [LY]
 H. B. Lawson and S.T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211228. MR 0334083 (48:12402)
 [L]
 O. Loos, Symmetric spaces, vol. 1, Benjamin, New York, 1969.
 [M]
 G. D. Mostow, Strong rigidity of locally symmetric spaces, Ann. of Math. Studies, No. 78, Princeton Univ. Press, Princeton, N.J., 1973. MR 0385004 (52:5874)
 [S]
 V. Schroeder, A splitting theorem for spaces of nonpositive curvature, Invent. Math. 79 (1985), 323327. MR 778131 (86b:53041)
 [W1]
 J. Wolf, Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 1418. MR 0163262 (29:565)
 [W2]
 , Space of constant curvature, 2nd ed., published by the author, Berkeley, 1972.
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DOI:
http://dx.doi.org/10.1090/S00029947198809570761
PII:
S 00029947(1988)09570761
Article copyright:
© Copyright 1988
American Mathematical Society
