On a class of twistorial maps

Details On a class of twistorial maps On a class of twistorial maps

2007-02-13

arxiv.org/abs/math/0702385v1

Mathematics

Differential Geometry

Preprint, I.M.A.R., 2006

Scientific paper

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.

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