Antiholomorphic curvature operator in the almost Hermitian geometry - Videv, Veselin (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 1996)
Let $(M,g,J)$ be $2n-$dimensional almost Hermitian manifold, $p$ be an arbitrary point of $M$, and $X,Y$ be an arbitrary orthonormal pair of tangent vectors in the tangent space $M_p$. If the plane $E^2(p;X,Y)$ is antiholomorphic, i.e. $E^2 \perp JE^2$, then we define the linear symmetric operator $α_{X,Y}:M_p → M_p$, where $α_{X,Y}(u)=\frac{1}{2}[R(u,X,Y)+R(u,Y,X)]$ In the present paper we consider the problem when the trace or the spectrum of the curvature operator $α_{X,Y}$ depends on the point $p \in M$ and not on the choice of $X \in M_p$