Twistors on Riemannian manifolds and CR-structures. (English. Russian original) Zbl 0779.53042

Alekseevskij, D. V.; Graev, M. M.

Russ. Math. 36, No. 5, 1-16 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 5(360), 3-19 (1992).

A CR-structure on a manifold \(M^ n\) is a distribution \(W\) of codimension \(k\) with a complex structure \(J\) on \(W\). In case the distribution \(W\) is given by the image \(W = Im({\mathcal C})\) of a skew- symmetric operator \(\mathcal C\) satisfying the equation \({\mathcal C}^ 3 + {\mathcal C} \equiv 0\) the CR-structure \((W,J = {\mathcal C}/W)\) on the Riemannian manifold \((M^ n,g)\) is called a Hermitian CR-structure. First of all the authors derive the integrability conditions for a Hermitian CR-structure using a connection preserving the Riemannian metric \(g\) as well as the operator \(\mathcal C\). In a natural way one can construct the twistor space \(Z\) of a Hermitian CR-structure. \(Z\) is a fibre bundle over \(M^ n\) with fibre \(SO(n)/U(m)\times SO(k)\) \((n = 2m+k)\), and admits again a Hermitian CR-structure. The CR-structure on \(Z\) is conformally invariant. The second result of the present paper contains the integrability conditions for the CR-structure on the twistor space. In particular, in case \(k = n-2\) the CR-structure on \(Z\) is always integrable. On the other hand, the CR-structure on the twistor space of type \(n\geq 5\), \(m\geq 2\) is integrable if and only if \((M^ n,g)\) is conformally flat. In the last part of the paper the authors discuss the case of a CR-structure of codimension \(k = 1\) and the integrability condition on the twistor space for a new CR-structure on \(Z\). It turns out that any Cartan connection determines a CR-structure on \(Z\) and this structure is integrable in case the torsion \(T\equiv 0\) vanishes.

Reviewer: Th.Friedrich (Berlin)

MSC:

53C55

Global differential geometry of Hermitian and Kählerian manifolds

Keywords:

Hermitian CR-structure; integrability conditions; conformally flat; Cartan connection