  We formulate the multi-band kp theory of hyperfine interactions for
semiconductor nanostructures in the envelope function approximation. We apply
this theoretical description to the fluctuations of the longitudinal and
transverse Overhauser field experienced by a hole for a range of InGaAs quantum
dots of various compositions and geometries. We find that for a wide range of
values of $d$-shell admixture to atomic states forming the top of the valence
band, the transverse Overhauser field caused by this admixture is of the same
order of magnitude as the longitudinal one, and band mixing adds only a minor
correction to this result. In consequence, the kp results are well reproduced
by a simple box model with the effective number of ions determined by the wave
function participation number, as long as the hole is confined in the
compositionally uniform volume of the dot, which holds in a wide range of
parameters, excluding very flat dots.
