A nullity condition for complex contact metric manifolds

Abstract

A nullity condition for real contact manifolds is defined by Blair, Koufogiorgos and Papantoniu. Lately, Boeckx classified such manifolds completely. In this paper, a nullity condition for complex contact manifolds is defined as follows: take a complex contact manifold whose vertical space is annihilated by the curvature. Then, apply an $\mathcal{H}$-homothetic deformation. In this way, we get a condition which is invariant under $\mathcal{H}$-homothetic deformations. A complex contact manifold satisfying this condition is called a complex (κ,μ)-space. Some curvature properties of complex (κ,μ)-spaces are studied and it is shown that, just as in the real case, the curvature tensor of a complex (κ,μ)-space is completely determined.