Harmonic maps and twistorial structures

We introduce the notion of Riemannian twistorial structure and we show that it provides new natural constructions of harmonic maps.Comment: 15 page

Twistor Theory for CR quaternionic manifolds and related structures

In a general and non metrical framework, we introduce the class of CR quaternionic manifolds containing the class of quaternionic manifolds, whilst in dimension three it particularizes to, essentially, give the conformal manifolds. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic manifolds.Comment: The paper has been split into two parts: 1. S. Marchiafava, L. Ornea, R. Pantilie, Twistor Theory for CR quaternionic manifolds and related structures; 2. S. Marchiafava, R. Pantilie, Twistor Theory for co-CR quaternionic manifolds and related structures. This is the first part. The second part will, also, be posted on the ArXiv; Monatshefte fuer Mathematik, 201

Multidimensional integrable systems and deformations of Lie algebra homomorphisms

We use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of anti--self--dual Yang--Mills equations with a gauge group Diff$(S^1)$.Comment: 14 pages. An example of a reduction to the Beltrami equation added. New title. Final version, published in JM

Trkalian fields: ray transforms and mini-twistors

We study X-ray and Divergent beam transforms of Trkalian fields and their relation with Radon transform. We make use of four basic mathematical methods of tomography due to Grangeat, Smith, Tuy and Gelfand-Goncharov for an integral geometric view on them. We also make use of direct approaches which provide a faster but restricted view of the geometry of these transforms. These reduce to well known geometric integral transforms on a sphere of the Radon or the spherical Curl transform in Moses eigenbasis, which are members of an analytic family of integral operators. We also discuss their inversion. The X-ray (also Divergent beam) transform of a Trkalian field is Trkalian. Also the Trkalian subclass of X-ray transforms yields Trkalian fields in the physical space. The Riesz potential of a Trkalian field is proportional to the field. Hence, the spherical mean of the X-ray (also Divergent beam) transform of a Trkalian field over all lines passing through a point yields the field at this point. The pivotal point is the simplification of an intricate quantity: Hilbert transform of the derivative of Radon transform for a Trkalian field in the Moses basis. We also define the X-ray transform of the Riesz potential (of order 2) and Biot-Savart integrals. Then, we discuss a mini-twistor respresentation, presenting a mini-twistor solution for the Trkalian fields equation. This is based on a time-harmonic reduction of wave equation to Helmholtz equation. A Trkalian field is given in terms of a null vector in C3 with an arbitrary function and an exponential factor resulting from this reduction.Comment: 37 pages, http://dx.doi.org/10.1063/1.482610

Twistor Theory for co-CR quaternionic manifolds and related structures

In a general and non metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kaehler manifolds

A note on CR-quaternionic maps

We introduce the notion of CR quaternionic map and we prove that any such real analytic map, between CR quaternionic manifolds, is the restriction of a quaternionic map between quaternionic manifolds. As an application we prove, for example, that for any submanifold M, of dimension 4k-1, of a quaternionic manifold N, such that TM generates a quaternionic subbundle of TN restricted to M, of (real) rank 4k, there exists, locally, a quaternionic submanifold of N, containing M as a hypersurface

Twistorial maps between quaternionic manifolds

We introduce a natural notion of quaternionic map between almost quaternionic manifolds and we prove the following, for maps of rank at least one A map between quaternionic manifolds endowed with the integrable almost twistorial structures is twistorial if and only if it is quaternionic A map between quaternionic manifolds endowed with the nonintegrable almost twistorial structures is twistorial if and only if it is quaternionic and totally-geodesic As an application, we describe all the quaternionic maps between open sets of quaternionic projective space