  A Riemannian manifold is called geometrically formal if the wedge product of
any two harmonic forms is again harmonic. We classify geometrically formal
compact 4-manifolds with nonnegative sectional curvature. If the sectional
curvature is strictly positive, the manifold must be homeomorphic to S^4 or
diffeomorphic to CP^2. This conclusion stills holds true if the sectional
curvature is strictly positive and we relax the condition of geometric
formality to the requirement that the length of harmonic 2-forms is not too
nonconstant. In particular, the Hopf conjecture on S^2 x S^2 holds in this
class of manifolds.
