Topological aspects of conformally flat manifolds (Q919331)

The author examines the topological structure of conformally flat manifolds according to their limit sets. A Riemannian closed n-manifold (M,g) is said to be conformally flat if for any \(x\in M\) there exist a neighbourhood U of x and a smooth embedding \(\phi: U\to S^ n\) (n- sphere) such that \(\phi^*g_ S=\mu g\), where \(g_ S\) is the spherical metric of \(S^ n\) and \(\mu\) is a positive valued continuous function on U. Further M is called elementary if its limit set is finite. For \(n\geq 3\), these manifolds are completely classified according to the cardinality of the limit set. Moreover the author constructs a conformally flat 3-manifold whose limit set is a wild Cantor set. This answers a question stated by \textit{M. Bestvina} and \textit{D. Cooper} in Proc. Am. Math. Soc. 99, 623-626 (1987; Zbl 0622.57030).