2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62626We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor $ψ$ of algebraic type $F \cdot ψ= 0$ with respect to the unique connection $\nabla$ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of almost contact metric manifolds and discuss a link between them and the dilation function in 5-dimensional string theory. We find natural conditions implying conformal invariances of parallel spinors. We present topological obstructions to the existence of parallel spinors in the compact case.Latex2e, 15 pages; Final version of the paper, the preliminary title has been "Almost contact manifolds and type II equations"Differential GeometryHigh Energy Physics - Theory53C25, 53C27, 53C55, 81T30text