A hyperbolic twistor space (Q2703559)

Almost quaternionic structures of the second kind are manifolds \(M\) endowed with a rank \(3\) subbundle of \(\text{End}(TM)\) locally generated by an almost complex structure \(J_1\) and two almost product structures \(J_2,J_3\), all of three anticommuting. The paraquaternionic projective spaces are but one example. When \(\dim M=4n\), a semi-Riemannian metric with signature \((2n,2n)\) can always be found with respect to which \(J_1\) acts as an isometry and \(J_2, J_3\) as anti-isometries. NEWLINENEWLINENEWLINEThis paper initiates the study of a twistor space associated to such a structure. The fibre is now modeled on the hyperbolic plane with a metric of constant sectional curvature \(-1\). As in the quaternionic case, two almost complex structures are defined on the total space and their integrability is discussed. But unlike the quaternionic case, where the total space of the twistor bundle cannot carry any non-constant holomorphic function, on the twistor space of \(\mathbb{R}^4\), endowed with its canonical quaternionic structure of the second kind, an example of a non-constant holomorphic function is constructed.