A structure on almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu, cosymplectic. We introduce semi-invariant -submersions from a manifold with such structure onto a Riemannian manifold. We give examples, investigate the geometry of foliations defined from a Riemannian submersion, and find necessary and sufficient conditions for a semi-invariant -submersion to be totally geodesic.

generalized quasi-Sasakian manifold, Riemannian submersion, semi-invariant -submersion.