On a class of twistorial maps

We show that a natural class of twistorial maps gives a pattern for apparently different geometric maps, such as, (1,1)(1,1)-geodesic immersions from (1,2)(1,2)-symplectic almost Hermitian manifolds and pseudo horizontally conformal submersions with totally geodesic fibres for which the associated almost CR-structure is integrable. Along the way, we construct for each constant curvature Riemannian manifold (M,g)(M,g), of dimension m  , a family of twistor spaces {Zr(M)}1⩽r<12m such that Zr(M)Zr(M) parametrizes naturally the set of pairs (P,J)(P,J), where P   is a totally geodesic submanifold of (M,g)(M,g), of codimension 2r, and J is an orthogonal complex structure on the normal bundle of P which is parallel with respect to the normal connection.