ar X iv : m at h / 05 02 32 5 v 1 [ m at h . D G ] 1 5 Fe b 20 05 ON NON - NEGATIVELY CURVED METRICS ON OPEN FIVE - DIMENSIONAL MANIFOLDS

Let V n be an open manifold of non-negative sectional curvature with a soul Σ of co-dimension two. The universal cover Ñ of the unit normal bundle N of the soul in such a manifold is isometric to the direct productM×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 → Σ on the factor M in this metric splitting Ñ = M × 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 ofM×(R, gF )×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 and Σ 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 × (R, gF )×R by the flow along corresponding Killing field.