  The valence band states confined in infinitely deep quantum dots made of
diluted magnetic semiconductors (DMS) are considered theoretically. A complex
anisotropic structure of the valence bands in DMSs with cubic symmetry
described by the full Luttinger Hamiltonian is taken into account. It is found
that the Zeeman splitting is very sensitive to the shape of the confining
potential and, in particular, to its orientation relative to the direction of
an external magnetic field. This sensitivity has its origin in a mixing of
different spin components of a hole wave function which takes place for finite
hole wave vectors ${\bf k}$. Several consequences of the effect are discussed,
including a possibility to control the inter-dot tunneling by an external
magnetic field. It is shown also that the polarizations of optical transitions
in a single DMS quantum dot depend on details of geometry of its confining
potential as well as on the strength of the magnetic field.
