  Exclusive electroproduction of $\omega$ mesons on unpolarized hydrogen and
deuterium targets is studied in the kinematic region of Q$^2$>1.0 GeV$^2$, 3.0
GeV < W < 6.3 GeV, and -t'< 0.2 GeV$^2$. Results on the angular distribution of
the $\omega$ meson, including its decay products, are presented. The data were
accumulated with the HERMES forward spectrometer during the 1996-2007 running
period using the 27.6 GeV longitudinally polarized electron or positron beam of
HERA. The determination of the virtual-photon longitudinal-to-transverse
cross-section ratio reveals that a considerable part of the cross section
arises from transversely polarized photons. Spin density matrix elements are
presented in projections of Q$^2$ or -t'. Violation of s-channel helicity
conservation is observed for some of these elements. A sizable contribution
from unnatural-parity-exchange amplitudes is found and the phase shift between
those amplitudes that describe transverse $\omega$ production by longitudinal
and transverse virtual photons, $\gamma^{*}_{L} \to \omega_{T}$ and
$\gamma^{*}_{T} \to \omega_{T}$, is determined for the first time. A hierarchy
of helicity amplitudes is established, which mainly means that the
unnatural-parity-exchange amplitude describing the $\gamma^*_T \to \omega_T$
transition dominates over the two natural-parity-exchange amplitudes describing
the $\gamma^*_L \to \omega_L$ and $\gamma^*_T \to \omega_T$ transitions, with
the latter two being of similar magnitude. Good agreement is found between the
HERMES proton data and results of a pQCD-inspired phenomenological model that
includes pion-pole contributions, which are of unnatural parity.
