The twistor theory of the Hermitian Hurwitz pair (ℂ⁴(I _2,2),ℝ(I _2,3))

Abstract

An analogue of the twistor theory is given for the Hermitian Hurwitz pair(ℂ⁴(I _2,2),ℝ(I _2,3)). In Sect. 2 a concept of Hurwitz twistors is introduced and a counterpart of the Penrose correspondence is obtained. It is proved that there exists a one-to-one correspondence between the twistors on the (1,3)-space and the (2,2)-space, which is called the duality theorem for Hurwitz twistors (Theorem 1). In Sect. 3. a concept of spinor equations is introduced for an Hermitian Hurwitz pair (abbreviated as HHP) and the duality theorem for solutions of the spinor equations is proved (Theorem 2). In Sect. 4 we give an elementary proof of the Penrose theory on the base of our Key Lemma. Then we can give the desired correspondence explicitly. In sect. 5 we consider the Penrose theory in the context of HHPs. At first we give a local version. It is proved that every solution of the spinor equation on the (2,2)-space can be represented as a ∂-harmonic one-form. By use of this result, we can get a direct relationship between the complex analysis and spinor theory on some open setM ⁺, which is called as “semi-global version” of the Penrose theory (Theorem 7). Moreover, we can get the original Penrose theory by use of the Penrose transformation (Theorem 5).