Rakić duality principle in the almost Hermitian geometry

Abstract

Rakic duality principle turns out to be one of the crucial steps in proving Osserman conjecture. Basically, it claims that if $${\mathcal{R}}$$ is an Osserman algebraic curvature tensor and X and Y are unit vectors, then Y is an eigenvector of the Jacobi operator $${\mathcal{R}(\cdot, X)X}$$ if and only if X is an unit eigenvector of $${\mathcal{R}(\cdot, Y)Y}$$ with the same eigenvalue. We prove necessary and sufficient conditions for certain almost Hermitian manifolds, the so called AH 3-manifolds, to have pointwise constant holomorphic curvature and pointwise constant antiholomorphic sectional curvature. It turns out that for this class of almost Hermitian manifolds these conditions are directly connected to the duality principle.