Geometrically formal 4-manifolds with nonnegative sectional curvature

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 $\mathbb{CP}^2$.

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 $S^2 \times S^2$ holds in this class of manifolds.

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Published 30 January 2015