On space-times with commutative and conformal anti-invariant pairing of null vector fields (Q2718054)

Let \((M,g)\) be a general space-time, and let \(S=\{h_A\mid A=1,2,3,4\}\) be a Sachs frame (or a null frame) over \(M\) (the complex vectorial formalism is used). Denote by \(h_1,h_4\) the real vector fields, and by \(h_2,h_3\) the complex conjugate ones. One assumes that the pairings \((h_1,h_4)\) and \((h_2,h_3)\) are commutative and conformal anti-invariant. Let \(Z^3\) denote the structure almost symplectic form of \(M\). The authors prove that the following properties are mutually equivalent: NEWLINENEWLINENEWLINE(i) The pairings \((h_1,h_4)\) and \((h_2,h_3)\) are null commutative and conformal anti-invariant. NEWLINENEWLINENEWLINE(ii) \(M\) is a Tachibana manifold. NEWLINENEWLINENEWLINE(iii) \(Z^3\) is a symplectic form which together with its complex conjugate form \(\overline{Z}^3\) defines an almost couple in the sense of Geiges, i.e., \(Z^3\wedge \overline{Z}^3=0\). NEWLINENEWLINENEWLINEIf in addition \(M\) belongs to type \(D\) in Petrov's classification, then \(M\) is of Schwarzschild type, and \(M\) reduces to a Minkowski manifold.