New structures on tangent bundles

Abstract

We construct three kinds of almost complex structures compatible with some naturally defined Riemannian metrics on the tangent bundle of an almost Hermitian manifold and in particular, of a complex space form. We use these structures to give a class of hyperhermitian structures and a one-parameter family of hyperkähler structures on the tangent bundle of a complex space form of positive holomorphic sectional curvature.