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 $M^{n-2}\times R$. 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 $π:V\toΣ$ on the factor $M$ in this metric splitting $\tilde N=M\times R$. The case $n=4$ was considered in [GT] 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 [GT] 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 $Σ^3$ 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.