The twistor theory of the Hermitian Hurwitz pair (ℂ4(I 2,2),ℝ(I 2,3))
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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).
1991 Mathematics Subject Classification32L25 Secondary 346C20
Key wordsClifford analysis Hurwitz pair twistor space with an indefinite scalar product
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