Certain 4-manifolds with non-negative sectional curvature

In this paper, we study certain compact 4-manifolds with non-negative sectional curvature K. If s is the scalar curvature and W(center dot) is the self-dual part of Weyl tensor, then it will be shown that there is no metric g on S(center dot) x S(center dot) with both (i) K > 0 and (ii) (sic)s - W(center dot) >= 0. We also investigate other aspects of 4-manifolds with non-negative sectional curvature. One of our results implies a theorem of Hamilton: "If a simply-connected, closed 4-manifold M(center dot) admits a metric g of non-negative curvature operator, then M(center dot) is one of S(center dot), CP(center dot) and S(center dot) x S(center dot)". Our method is different from Hamilton's and is much simpler. A new version of the second variational formula for minimal surfaces in 4-manifolds is proved.