Conformally flat manifolds whose development maps are not surjective. I.

*(English)*Zbl 0608.53036A 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.

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.) |

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\textit{Y. Kamishima}, Trans. Am. Math. Soc. 294, 607--623 (1986; Zbl 0608.53036)

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