Antiholomorphic curvature operator in the almost Hermitian geometry

Abstract

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. $E^2\perp J{E}^2$, then we define the linear symmetric operator ${\alpha }_{X,Y}:M_p\to M_p$, where
${\alpha }_{X,Y}(u)=\frac{1}{2}[R(u,X,Y)+R(u,Y,X)]$

In the present paper we consider the problem when the trace or the spectrum of the curvature operator ${\alpha }_{X,Y}$ depends on the point $p\in M$ and not on the choice of $X\in M_p$