Non - Negatively Curved Metrics on S 2 × S 2 Admitting Killing Fields

A metric of non-negative sectional curvature on the direct product S2 × S2 of two two-spheres may be introduced as follows: take spheres (S2 i , gi), i = 1, 2 with rotationally symmetric metrics of non-negative curvature and consider the factor space of (S2, g1) × (S2, g2) × S1 by the action of the family of isometries Ĩτ (φ1, φ2) acting as the composition of a rotation of the first factor by an angle φ1τ with a rotation of the second factor by an angle φ2τ and a rotation of the third factor S1 by an angle τ for some 0 ≤ φ1, φ2 < 2π fixed (when some φi equals zero the corresponding factor (S2, gi) is not assumed rotationally symmetric). The obtained factor space (N4, g) is diffeomorphic to the direct product S2 × S2, and its metric has non-negative sectional curvature (by the O’Neill’s theorem). This metric is not isometric to the direct product metric (except when some φi equals zero). Such factor space (N4, g) inherits from (S2, g1)× (S2, g2)× S1 the one-parameter family of Killing vector fields generated by other isometries Ĩτ (φ1, φ ′ 2), since they commute with Ĩτ (φ1, φ2) (again, except the case when some φi equals zero when all such Killing fields become proportional to each other). We consider an arbitrary metric g of non-negative sectional curvature on S2×S2 and prove that it is isometric to some (N4, g) as above if it admits a non-trivial Killing field. In particular, the obtained result affirms the ”variational” Hopf conjecture (via Bourguignon, Deschamps, Sentenac theorem): the non-existence of analytical variations (S2 × S2, g(t)) of the product metric with strictly positive curvature for t > 0. A positively curved closed (i.e., compact without boundary) four-manifold N is homeomorphic to S or CP 2 if it admits a non-trivial Killing vector field X , see [HK]. Non-negatively curved four-manifolds and not homeomorphic to S or CP 2 may well have non-trivial Killing fields. For instance, take two spheres (S i , gi), i = 1, 2 with rotationally symmetric metrics of non-negative curvature and consider the factor space of (S, g1) × (S, g2)× S by the R-action of the family of isometries Ĩτ (φ1, φ2) acting as the composition of a rotation of the first factor by an angle φ1τ with a rotation of the second factor by an angle φ2τ and a rotation of the third factor S by an angle τ for some 0 ≤ φ1, φ2 < 2π fixed (when some φi equals zero the corresponding factor (S, gi) is not assumed rotationally symmetric). The obtained factor space (N, g) is diffeomorphic to the direct product S × S, and its metric is of non-negative sectional curvature (by the O’Neill’s theorem). This metric is not isometric to the direct product metric except when some φi equals zero. Such factor space (N , g) inherits from (S, g1)× (S, g2)×S the one-parameter family of Killing vector fields generated by other isometries Ĩτ (φ1, φ2), since they commute with Ĩτ (φ1, φ2) (again, except the case when some φi equals zero when all such Killing fields 1991 Mathematics Subject Classification. 53C20, 53C21. Supported by the Faculty of Natural Sciences of the Hogskolan i Kalmar, (Sweden). Submitted October 31, 2005; revised February 20, 2006..