Y. Matsushima proved that in a compact Kähler-Einstein space, any contravariant analytic vector field is uniquely decomposed in the form of Killing vector fields. A. Lichnerowicz generalised this result for a compact Kähler manifold with constant scalar curvature. S. Sawaki generalised the same result for a compact almost Tachibana-Einstein manifold. In 1959, Tachibana introduced the concept of an almost analytic vector field in the almost complex spaces.

The aim of the paper is to discuss an analytic vector field on an almost hyperbolic Hermitian manifold and to obtain certain expressions of the curvature tensor on an almost hyperbolic Hermitian manifold admitting semi-symmetric metric connection assuming analytic vector field to be killing and affine.