Rigidity for nonnegatively curved metrics on \(S^2 \times \mathbb R^3\) (Q1424732)

In [Geom. Dedicata 99, 127--136 (2003; Zbl 1032.53030)] \textit{D. Gromoll} and the author show that the space \(S^ 2\times \mathbb R^ 2\) equipped with an arbitrary complete metric of nonnegative curvature is a product metric or is isometric to a Riemannian product \(((S^ 2,g_ 0)\times (\mathbb R^ 2, g_ f)\times \mathbb R)/\mathbb R\), where \(\mathbb R\) acts diagonally on the product by the flow along Killing fields on \((S^ 2,g_ 0)\) and \((\mathbb R^ 2,g_ f)\) and by translations on \(\mathbb R\). The question whether \(S^ 2\times S^ 2\) admits positive or quasi-positive curvature is a long-standing unsolved problem in Riemannian geometry. In this paper, the author studies metrics of non-negative curvature on \(S^ 2\times \mathbb R^ 3\). First, he classifies the connection metrics and gives several examples of nonconnection metrics. Finally, for metrics more general than connection metrics the author demonstrates the case under which the inequality rigidly determines what metrics may look like at soul. This provides the evidence that the family of metrics is small.