Harmonic Maps and Twistorial Structures

Abstract

We introduce the notion of Riemannian twistorial structure and show that it provides new natural constructions of harmonic maps.

Almost $G_2$-manifolds with almost twistorial structures

“…The main source of Euclidean twistorial structures [3] is provided by the closed orbits, on the Grassmannians of (co)isotropic spaces, of the complexification of a compact Lie group, endowed with a faithfull orthogonal representation. For G 2 , there are two 'fundamental' such (generalized) Grassmannians: the hyperquadric Q in the projectivisation of the space U of complex imaginary octonions, and the space Y of anti-self-dual spaces in U .…”

“…[5] ). As the dual of this is not maximal (in the sense of [3] ) we are led to, also, consider an Euclidean twistorial structure on C × U . See Section 2 , where the integrability of the obtained canonical almost twistorial structures is studied, by using [1] .…”

Twistor theory for exceptional holonomy

“…In the joint work [4] , we introduced the notion of 'Riemannian twistorial structure' as the necessary augmentation that makes the Riemannian manifolds the objects of a category. Furthermore, we have shown that such structures can be found on any simplyconnected Riemannian symmetric space.…”

Twistor Theory for Exceptional Holonomy

“…(i) There exists a so(7)-invariant diffeomorphism from Gr 0 3 (7) onto Q 6 . (ii) There exists a g 2 -invariant embedding of Q 5 into Gr 0 3 (7). (iii) There exists an embedding of Q 5 into Gr 0 4 (8), which is equivariant with respect to the embedding morphism from g 2 into so(7).…”

“…Example 2.5. Let Z be the projectivization of the dual of the tautological bundle over Q 5 ⊆ Gr 0 4 (8), endowed with the images of the sections of the projection π : Z → Q 5 . An open neighborhood of each such quadric is the twistor space of a flat G 2 -manifold.…”