Null-geodesics in complex conformal manifolds and the LeBrun correspondence (Q2729301)

The main result of the paper states that a conformal complex manifold containing a compact, simply connected, null geodesic is conformally flat. The case of a complex projective manifold was studied by \textit{Y.-G. Ye} [Int. J. Math. 5, 141-168 (1994; Zbl 0802.53021)]. The most difficult case is that of dimension \(4\), which uses the LeBrun correspondence [\textit{C. R. LeBrun}, Proc. R. Soc. Lond., Ser. A 380, 171-185 (1982; Zbl 0549.53042) and Trans. Am. Math. Soc. 284, 601-616 (1984; Zbl 0513.53006)]. A relation between the two conformal invariants of the manifolds involved in the LeBrun correspondence, or, more generally, of an umbilic submanifold of dimension \(3\) and of its self-dual ambient space of dimension \(4\), is also obtained.