Orthogonal almost complex structure and its Nijenhuis tensor

Abstract In this paper, we demonstrate that on an almost Hermitian manifold , a 2‐form , the pullback of the Kähler form on the twistor bundle over , is nondegenerate if the squared norm of the Nijenhuis tensor is less than when or less than 16 when . As one of the consequences, there exists no orthogonal almost complex structure on the standard sphere with everywhere.