  We show that a natural class of twistorial maps gives a pattern for
apparently different geometric maps, such as, $(1,1)$-geodesic immersions from
$(1,2)$-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 $(M,g)$, of dimension $m$, a family of twistor
spaces $\bigl\{Z_r(M)\bigr\}_{1\leq r<\tfrac12m}$ such that $Z_r(M)$
parametrizes naturally the set of pairs $(P,J)$, where $P$ is a totally
geodesic submanifold of $(M,g)$, 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.
