  In this paper, we study certain compact 4-manifolds with non-negative
sectional curvature $K$. If $s$ is the scalar curvature and $W_+$ is the
self-dual part of Weyl tensor, then it will be shown that there is no metric
$g$ on $S^2 \times S^2$ with both (i) $K > 0$ and (ii) $ {1/6} s - W_+ \ge 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 $M^4$ admits a metric $g$ of non-negative
curvature operator, then $M^4$ is one of $S^4$, $\Bbb CP^2$ and $S^2 \times
S^2$".
  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.
