Rigidity of Non-negatively Curved Metrics on Open Five-dimensional Manifolds

As the first step in the direction of the Hopf conjecture on the non-existence of metrics with positive sectional curvature on S2 × S2 the authors of [GT] suggested the following (Weak Hopf) conjecture (on the rigidity of non-negatively curved metrics on S2 × R3): ”The boundary S2 × S2 of the S2 × B3 ⊂ S2 × R3 with an arbitrary complete metric of non-negative sectional curvature contains a point where a curvature of S2 × S2 vanish”. In this note we verify this. More ”flats” in M Let (M, g) be a complete open Riemannian manifold of non-negative sectional curvature. Remind that as follows from [CG] and [P] an arbitrary complete open manifold M of non-negative sectional curvature contains a closed absolutely convex and totally geodesic submanifold Σ (called a soul) such that the projection π : M → Σ of M onto Σ along geodesics normal to Σ is well-defined and is a Riemannian submersion. The (vertical) fibers WP = π (P ), P ∈ Σ of π define a metric foliation in M and two distributions: a vertical V distribution of subspaces tangent to fibers and a horizontal distribution H of subspaces normal to V . For an arbitrary point P on Σ, an arbitrary geodesic γ(t) on Σ and arbitrary vector field V (t) which is parallel along γ and normal to Σ the following (1) Π(t, s) = expγ(t)sV (t) are totally geodesic surfaces in M of zero curvature, i.e., flats. Sometimes, these are the only directions of zero curvature in open M (e.g., when M is the tangent bundle to the two-dimensional sphere with the Cheeger-Gromoll metric, see [M2]). The objective of this note is to verify the (Weak Hopf) conjecture from [GT] and to point to more directions of zero curvature in our particular case of a five-dimensional M . The following statement is true. 1991 Mathematics Subject Classification. 53C20, 53C21. Supported by the Swedish Science Consul (Vetenskapr̊adet) and the Faculty of Natural Sciences of the Hogskolan i Kalmar, (Sweden). Submitted August 12, 2004; revised November 29, 2004.