In this short note, we analyze geometric properties of orbit spaces of certain involutions in dimensions four, five, and six. We consider constructions of F-structures on manifolds of dimension at least four that allows us to study minimal entropy, minimal volume, collapse with bounded curvature, and sign of the Yamabe invariant, and its vanishing, building on work of Paternain–Petean. The existence of Riemannian metrics of vanishing topological entropy on the orbit spaces is investigated as well
