Harmonic morphisms between Weyl spaces and twistorial maps II

Abstract

We define, on smooth manifolds, the notions of almost twistorial structure and twistorial map, thus providing a unified framework for all known examples of twistor spaces. The condition of being harmonic morphisms naturally appears among the geometric properties of submersive twistorial maps between low-dimensional Weyl spaces endowed with a nonintegrable almost twistorial structure due to Eells and Salamon. This leads to the twistorial characterisation of harmonic morphisms between Weyl spaces of dimensions four and three. Also, we give a thorough description of the twistorial maps with one-dimensional fibres from four-dimensional Weyl spaces endowed with the almost twistorial structure of Eells and Salamon.

Harmonic Maps and Twistorial Structures

“…This brings us to the other main task of this paper, namely, to find new natural constructions of harmonic maps. On the one hand, this is in continuation of our investigation of the interplay of the properties of a map of being twistorial and a harmonic morphism , and, on the other hand, it provides the adequate level of generality for previously known twistorial constructions of harmonic maps . We achieve this just because, in our setting, any simply connected Riemannian symmetric space admits a nontrivial Riemannian twistorial structure invariant under the isometry group (Theorem ) leading, for example, to the natural twistorial constructions of Example , based on the new twistorial structure of Example (3).…”

“…Then is an almost CR twistor space (cf. ) of M if there is a proper surjective submersion as follows.…”

“…The following definition specializes notions of ; compare, also, . Definition If the torsion of ∇ is totally antisymmetric, we say that is a Riemannian almost twistorial structure (on M ) .…”

“…Furthermore, if σ is a Hermitian Y ‐structure, then the radial projection from onto the unit sphere induces a Riemannian twistorial structure on S , with ∇ the restriction to S of the trivial connection on U . The corresponding twistor space is the complex projectivization , and note that the radial projection is the twistorial map corresponding to the holomorphic submersion .…”

“…[20]). Note that this is not too restrictive as, assuming the complexified representation irreducible, we can, for example, take Y to be the closed G C -orbit in the projectivization of the complexification of U * , setting which proved to be crucial in the classification of irreducible torsion-free affine holonomies [14,15].Consequently, the "G-structures" should be given by suitable twistorial structures and the "morphisms" be just the corresponding twistorial maps of [11,22].One of the main tasks of this paper is to construct the suitable definition of "Riemannian twistorial structure." This is based on the "Euclidean (twistorial) Y -structures," that is, embeddings of Y into the generalized Grassmannian of coisotropic subspaces (that is, orthogonal complements of isotropic subspaces) of the corresponding Euclidean space.…”