  We discuss resonances for a nonrelativistic and spinless quantum particle
confined to a two- or three-dimensional Riemannian manifold to which a finite
number of semiinfinite leads is attached. Resolvent and scattering resonances
are shown to coincide in this situation. Next we consider the resonances
together with embedded eigenvalues and ask about the high-energy asymptotics of
such a family. For the case when all the halflines are attached at a single
point we prove that all resonances are in the momentum plane confined to a
strip parallel to the real axis, in contrast to the analogous asymptotics in
some metric quantum graphs; we illustrate it on several simple examples. On the
other hand, the resonance behaviour can be influenced by a magnetic field. We
provide an example of such a `hedgehog' manifold at which a suitable
Aharonov-Bohm flux leads to absence of any true resonance, i.e. that
corresponding to a pole outside the real axis.
