# zbMATH — the first resource for mathematics

Conformally flat manifolds whose development maps are not surjective. I. (English) Zbl 0608.53036
A Riemannian manifold M is conformally flat if for each point m of M there exist open sets $$U\subseteq M$$ and $$O\subseteq R^ n$$ and a conformal diffeomorpism $$x: O\to U$$ with $$m\in U=x(O)$$. For conformally flat manifolds one has the following uniformization theorem that follows from work of N. Kuiper and R. Kulkarni: Theorem. Let M be a smooth manifold of dimension $$n\geq 3$$, and regard M as a quotient space $$X/\pi$$ where X is the universal Riemannian cover of M and $$\pi \subseteq Isom(X)$$ is the fundamental group of M. If M is conformally flat, then there exists a conformal immersion $$d: X\to S^ n$$ and a homomorphism $$\phi: \pi\to Conf(S^ n)$$ such that $$\phi (\gamma)\circ d=d\circ \gamma$$ for all elements $$\gamma$$ of $$\pi$$. The pair $$(d,\phi)$$ are unique up to conjugation by an element of $$Conf(S^ n)$$, the group of conformal transformations of the sphere $$S^ n.$$
The group $$\phi(\pi) \subseteq Conf(S^ n)$$ is called the holonomy group of M. The group $$Conf(S^ n)$$ may be identified with $$SO(1,n+1)$$, the connected group of isometries of the $$n+1$$ dimensional hyperbolic space $$H^{n+1}$$. In fact, if one regards $$H^{n+1}$$ as the open unit ball in $$R^{n+1}$$ together with a suitable metric (conformally equivalent to the Euclidean metric), then the isometries of $$H^{n+1}$$ extend uniquely to conformal maps of $$S^ n$$ and vice versa, where $$S^ n$$ is regarded both as the topological boundary of $$H^{n+1}$$ in $$R^{n+1}$$ and as the geometric boundary $$H^{n+1}(\infty)$$ consisting of equivalence classes of asymptotic geodesics of $$H^{n+1}$$. Now define $$\partial (dX)$$ to be $$\overline{dX}-dX$$, where dX denotes the image of X in $$S^ n$$ under the conformal immersion d above. By construction the set $$\partial (dX)$$ is closed and invariant under the holonomy group $$\Gamma =\phi (\pi)$$. If $$\partial (dX)$$ is a single point, then M is flat and $$\phi$$ : $$\Gamma \to \pi$$ is an isomorphism by a result of Fried. Otherwise $$\partial (dX)$$ lies in the hyperbolic limit set $$L(\Gamma)\subset H^{n+1}(\infty)$$ and equals $$L(\Gamma)$$ if $$\partial (dX)$$ has at least 3 points.
Using facts from hyperbolic geometry about $$L(\Gamma)$$ the author obtains the following results. Theorem A. Let M be a 3-dimensional closed conformally flat manifold whose development map $$d: X\to S^ n$$ is not surjective. Then either the holonomy group is discrete or M is a locally homogeneous Riemannian manifold of nonpositive sectional curvature. In the latter case some finite covering of M is diffeomorphic to a product $$S^ 1\times N_ g$$, where $$N_ g$$ denotes a compact surface of genus $$g\geq 2$$. Theorem B. Let M be a closed conformally flat manifold of dimension $$n\geq 3$$. If the development map $$d: X\to S^ n$$ is not surjective, then d is a covering map. Theorem C. Let M be a closed conformally flat manifold of dimension $$n\geq 3$$. If the holonomy group contains a solvable subgroup of finite index, then M is a spherical space form, a Euclidean space form or a non-negatively curved manifold finitely covered by a Hopf manifold $$S^ 1\times S^{n-1}$$. Theorem C is related to work of W. Goldman.
Reviewer: P.Eberlein

##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
Full Text:
##### References:
  R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1 – 49. · Zbl 0191.52002  Su Shing Chen and Patrick Eberlein, Isometry groups of simply connected manifolds of nonpositive curvature, Illinois J. Math. 24 (1980), no. 1, 73 – 103. · Zbl 0413.53029  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. · Zbl 0295.53023  P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45 – 109. · Zbl 0264.53026  David Fried, Closed similarity manifolds, Comment. Math. Helv. 55 (1980), no. 4, 576 – 582. · Zbl 0455.57005 · doi:10.1007/BF02566707 · doi.org  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. · Zbl 0518.53041  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. · Zbl 0237.53037  Yoshinobu Kamishima, Lorentz space forms and virtually solvable groups, Indiana Univ. Math. J. 34 (1985), no. 2, 249 – 258. · Zbl 0569.53027 · doi:10.1512/iumj.1985.34.34015 · doi.org  Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. · Zbl 0246.53031  N. H. Kuiper, On conformally-flat spaces in the large, Ann. of Math. (2) 50 (1949), 916 – 924. · Zbl 0041.09303 · doi:10.2307/1969587 · doi.org  N. H. Kuiper, On compact conformally Euclidean spaces of dimension >2, Ann. of Math. (2) 52 (1950), 478 – 490. · Zbl 0039.17701 · doi:10.2307/1969480 · doi.org  R. S. Kulkarni, Groups with domains of discontinuity, Math. Ann. 237 (1978), no. 3, 253 – 272. · Zbl 0369.20028 · doi:10.1007/BF01420180 · doi.org  R. S. Kulkarni, On the principle of uniformization, J. Differential Geom. 13 (1978), no. 1, 109 – 138. · Zbl 0381.53023  Ravi S. Kulkarni and Frank Raymond, 3-dimensional Lorentz space-forms and Seifert fiber spaces, J. Differential Geom. 21 (1985), no. 2, 231 – 268. · Zbl 0563.57004  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. · Zbl 0094.24501  H. Blaine Lawson Jr. and Shing Tung Yau, Compact manifolds of nonpositive curvature, J. Differential Geometry 7 (1972), 211 – 228. · Zbl 0266.53035  Michael L. Mihalik, Ends of double extension groups, Topology 25 (1986), no. 1, 45 – 53. · Zbl 0589.57001 · doi:10.1016/0040-9383(86)90004-2 · doi.org  Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401 – 487. · Zbl 0561.57001 · doi:10.1112/blms/15.5.401 · doi.org  Peter Scott, There are no fake Seifert fibre spaces with infinite \?$$_{1}$$, Ann. of Math. (2) 117 (1983), no. 1, 35 – 70. · Zbl 0516.57006 · doi:10.2307/2006970 · doi.org  John Stallings, Group theory and three-dimensional manifolds, Yale University Press, New Haven, Conn.-London, 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969; Yale Mathematical Monographs, 4. · Zbl 0241.57001  Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. · Zbl 0162.53304  Hitoshi Takagi, Conformally flat Riemannian manifolds admitting a transitive group of isometries, Tôhoku Math. J. (2) 27 (1975), no. 1, 103 – 110. , https://doi.org/10.2748/tmj/1178241040 Hitoshi Takagi, Conformally flat Riemannian manifolds admitting a transitive group of isometries. II, Tôhoku Math. J. (2) 27 (1975), no. 3, 445 – 451. · Zbl 0323.53037 · doi:10.2748/tmj/1203529254 · doi.org  Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171 – 202. · Zbl 0439.30034  Lipman Bers, Uniformization, moduli, and Kleinian groups, Bull. London Math. Soc. 4 (1972), 257 – 300. · Zbl 0257.32012 · doi:10.1112/blms/4.3.257 · doi.org  Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. · Zbl 0246.57017  William M. Goldman, Projective structures with Fuchsian holonomy, J. Differential Geom. 25 (1987), no. 3, 297 – 326. · Zbl 0595.57012  W. Thurston, The geometry and topology of $$3$$-manifolds, Princeton Univ. Press, Princeton, N.J., 1979.  Y. Kamishima, Conformal transformations and closed conformally flat $$3$$-manifolds. II, preprint, 1985.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.