  A periodic Schr\"odinger operator on a noncompact Riemannian manifold $M$
such that $H^1(M, \mathbb R)=0$ endowed with a properly discontinuous cocompact
isometric action of a discrete group is considered. Under some additional
conditions on the magnetic field existence of an arbitrary large number of gaps
in the spectrum of such an operator in the semiclassical limit is established.
The proofs are based on the study of the tunneling effect in the corresponding
quantum system.
