On non-negatively curved metrics on open five-dimensional manifolds

Abstract

Let V ⁿ be an open manifold of non-negative sectional curvature with a soul Σ of co-dimension two. The universal cover $\tilde N$ of the unit normal bundle N of the soul in such a manifold is isometric to the direct product M ^n-2 × R. In the study of the metric structure of V ⁿ an important role plays the vector field X which belongs to the projection of the vertical planes distribution of the Riemannian submersion $\pi\!\!:V\to\Sigma$ on the factor M in this metric splitting $\tilde N = M\times R$. The case n = 4 was considered in [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] where the authors prove that X is a Killing vector field while the manifold V ⁴ is isometric to the quotient of $M^2\times (R^2,g_F)\times R$ by the flow along the corresponding Killing field. Following an approach of [Gromoll, D., Tapp, K.: Geom. Dedicata 99, 127–136 (2003)] we consider the next case n = 5 and obtain the same result under the assumption that the set of zeros of X is not empty. Under this assumption we prove that both M ³ and Σ³ admit an open-book decomposition with a bending which is a closed geodesic and pages which are totally geodesic two-spheres, the vector field X is Killing, while the whole manifold V ⁵ is isometric to the quotient of $M^3\times (R^2,g_F)\times R$ by the flow along corresponding Killing field.