On complete metrics of nonnegative curvature on 2-plane bundles
Let \(K\) be a sectional curvature. \textit{J. Cheeger} and \textit{D. Gromoll} [Ann. of Math., II. Ser. 96, 413-443 (1972; Zbl 0246.53049)] proposed the following problem: Which vector bundles over closed manifolds with \(K\geq 0\) carry complete metrics with \(K\geq 0\)? The main purpose of this paper is to show that every 2-plane bundle and every \(S^1\)-bundle over the connected sum \(\mathbb{C} P^n\# \overline{\mathbb{C} P}^n\) of \(\mathbb{C} P^n\) with a negative \(\overline{\mathbb{C} P}^n\) carries a 2-parameter family of complete metrics with \(K\geq 0\). The author also solves this problem for a class of 2-plane bundles and principal \(S^1\)-bundles over \(M\), where \(M\) is an associated \(S^2\)-bundle over \(B\); \((B,h)\) is a Hodge manifold with positive Kählerian curvature. To construct the corresponding families of metrics he needs some properties of an Euler class in \(\widetilde{H}^2(M)\).
scientific article; zbMATH DE number 912161