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On a class of twistorial maps. (English) Zbl 1149.53028
Summary: 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 $$\{Z_r (M)\}_{1\leqslant r< \frac {1}{2} m}$$ 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.

MSC:
 53C28 Twistor methods in differential geometry 53C43 Differential geometric aspects of harmonic maps
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References:
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