Abstract
We prove that L 2 harmonic two-forms are parallel if a complete manifold (M, g) has the non-negative isotropic curvature. Furthermore, if (M, g) has positive isotropic curvature at some point, then there is no non-trivial L 2 harmonic two-form. We obtain that an almost Kahler manifold of non-negative isotropic curvature is Kahler and a symplectic manifold can not admit any almost Kahler structure of positive isotropic curvature.
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