On the topology of nonnegatively curved simply connected 4-manifolds with continuous symmetry
By discussing a conjecture of \textit{W. Y. Hsiang} and \textit{B. Kleiner} [J. Differ. Geom. 29, No. 3, 615-621 (1989; Zbl 0674.53047)] the authors study the topological classification of some nonnegatively curved 4- dimensional Riemannian manifolds. The following theorem is established: Theorem 1. Let \(M\) be a compact, simply connected, nonnegatively curved 4-dimensional Riemannian manifold with a nontrivial Killing vector field. Then \(M\) is homeomorphic to either \(S^ 4\), \(\mathbb{C} P^ 2\), \(S^ 2\times S^ 2\) or \(\mathbb{C} P^ 2\#\pm\mathbb{C} P^ 2\). If there is an effective isometric \(S^ 1\)-action on \(M\) (\(M\) is called an \(S^ 1\)- Riemannian manifold) such that its fixed point set \(F(S^ 1,M)\) contains two 2-dimensional connected components, then \(M\) is diffeomorphic to \(S^ 2\times S^ 2\) or \(-\mathbb{C} P^ 2\#\mathbb{C} P^ 2\). The second result of the paper is a double soul theorem for a complete nonnegatively curved \(n\)-dimensional \(S^ 1\)-Riemannian manifold.
scientific article; zbMATH DE number 624867
scientific article