Conformally flat manifolds whose development maps are not surjective. I

Kamishima, Yoshinobu

doi:10.1090/S0002-9947-1986-0825725-2

Conformally flat manifolds whose development maps are not surjective. I
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Trans. Amer. Math. Soc. 294 (1986), 607-623 Request permission

Abstract:

Let $M$ be an $n$-dimensional conformally flat manifold. A universal covering of $M, \tilde M$ admits a conformal development map into ${S^n}$. When a development map is not surjective, we can relate the boundary of the development image with the limit set of the holonomy group of $M$. In this paper, we study properties of closed conformally flat manifolds whose development maps are not surjective.

References

R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
Su Shing Chen and Patrick Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980), no. 1, 73–103. MR 550653
S. S. Chen and L. Greenberg, Hyperbolic spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 49–87. MR 0377765
P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648
David Fried, Closed similarity manifolds, Comment. Math. Helv. 55 (1980), no. 4, 576–582. MR 604714, DOI 10.1007/BF02566707
William M. Goldman, Conformally flat manifolds with nilpotent holonomy and the uniformization problem for $3$-manifolds, Trans. Amer. Math. Soc. 278 (1983), no. 2, 573–583. MR 701512, DOI 10.1090/S0002-9947-1983-0701512-8
Detlef Gromoll and Joseph A. 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), 545–552. MR 281122, DOI 10.1090/S0002-9904-1971-12747-7
Yoshinobu Kamishima, Lorentz space forms and virtually solvable groups, Indiana Univ. Math. J. 34 (1985), no. 2, 249–258. MR 783914, DOI 10.1512/iumj.1985.34.34015
Shoshichi Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972. MR 0355886
N. H. Kuiper, On conformally-flat spaces in the large, Ann. of Math. (2) 50 (1949), 916–924. MR 31310, DOI 10.2307/1969587
N. H. Kuiper, On compact conformally Euclidean spaces of dimension $>2$, Ann. of Math. (2) 52 (1950), 478–490. MR 37575, DOI 10.2307/1969480
R. S. Kulkarni, Groups with domains of discontinuity, Math. Ann. 237 (1978), no. 3, 253–272. MR 508756, DOI 10.1007/BF01420180
R. S. Kulkarni, On the principle of uniformization, J. Differential Geometry 13 (1978), no. 1, 109–138. MR 520605
Ravi S. Kulkarni and Frank Raymond, $3$-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), no. 2, 231–268. MR 816671
A. G. Kurosh, The theory of groups, Chelsea Publishing Co., New York, 1960. Translated from the Russian and edited by K. A. Hirsch. 2nd English ed. 2 volumes. MR 0109842
H. Blaine Lawson Jr. and Shing Tung Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211–228. MR 334083
Michael L. Mihalik, Ends of double extension groups, Topology 25 (1986), no. 1, 45–53. MR 836723, DOI 10.1016/0040-9383(86)90004-2
Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401–487. MR 705527, DOI 10.1112/blms/15.5.401
Peter Scott, There are no fake Seifert fibre spaces with infinite $\pi _{1}$, Ann. of Math. (2) 117 (1983), no. 1, 35–70. MR 683801, DOI 10.2307/2006970
John Stallings, Group theory and three-dimensional manifolds, Yale Mathematical Monographs, vol. 4, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. MR 0415622
Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740
Hitoshi Takagi, Conformally flat Riemannian manifolds admitting a transitive group of isometries, Tohoku Math. J. (2) 27 (1975), no. 1, 103–110. MR 442852, DOI 10.2748/tmj/1178241040
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586
Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257–300. MR 348097, DOI 10.1112/blms/4.3.257
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
William M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297–326. MR 882826

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