Algebraic conditions for compatibility of two metric forms with the same almost complex (quaternion) structure on a manifold (Q1603069)

Let \((M^{2n},g,J)\) and \((M^{2n},g',J)\) be almost Hermitian manifolds with the same almost-complex structure \(J\). In this paper the authors establish the block-diagonal structure of the matrix of the fundamental form of the Hermitian manifold \((M^{2n},g)\) with respect to the canonical skew-frame. The authors consider also the case of quaternionic manifolds \((M^{4m},g)\) and \((M^{4m},g')\) with equal quaternionic structure \(J_1,J_2,J_3\).