  We consider an almost complex manifold with Norden metric (i. e. a metric
with respect to which the almost complex structure is an anti-isometry). On
such a manifold we study a linear connection preserving the almost complex
structure and the metric and having a totally skew symmetric torsion tensor (i.
e. a 3-form). We prove that if a non-Kaehler almost complex manifold with
Norden metric admits such connection then the manifold is quasi-Kaehlerian (i.
e. has non-integrable almost complex structure). We prove that this connection
is unique, determine its form, and construct an example of it on a Lie group.
We consider the case when the manifold admits a connection with parallel
totally skew-symmetric torsion and the case when such connection has a Kaehler
curvature tensor. We get necessary and sufficient conditions for an isotropic
Kaehler manifold with Norden metric.
