On the decomposition of conformally flat manifolds
Let us call a manifold \(M\) conformally flat, if \(M\) admits a flat conformal structure. A well-known theorem of Kulkarni states that a connected sum of conformally flat manifolds is also conformally flat. A manifold \(M\) is said to be non-trivial if \(M\) is not diffeomorphic to \(S^ n\). And a compact non-trivial manifold \(M\) is called \(C\)-prime if \(M\) is conformally flat and \(M\) is not diffeomorphic to a connected sum of non-trivial conformally flat manifolds. It is proved in the present paper that any compact conformally flat manifold can be decomposed into a connected sum of a finite number of \(C\)-prime manifolds. Thus the classification problem of compact conformally flat manifolds is reduced to the classification of \(C\)-prime manifolds. Some sufficient conditions for a manifold to be \(C\)-prime and basic properties of \(C\)-prime manifolds are shown in the paper. Main results relate to the question how the scalar curvature (more precisely, a certain invariant concerning the Yamabe invariant of a flat conformal structure) changes, if we take a connected sum of conformally flat manifolds or if we decompose a conformally flat manifold into a connected sum of conformally flat manifolds. As an application of these results, the classification of compact oriented 3-manifolds admitting a conformally flat metric with positive scalar curvature is given.
scientific article; zbMATH DE number 469054