  A local classification of the Hermitian manifolds with flat associated
connection is given. Hermitian manifolds admitting locally a conformal metric
with flat associated connection are characterized by a curvature identity.
Locally conformal Kaehler manifolds as well as Hermitian surfaces with
vanishing associated conformal curvature tensor are characterized as locally
conformal to a Kaehler manifold of constant holomorphic sectional curvatures.
