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Twistors on Riemannian manifolds and CR-structures. (English. Russian original) Zbl 0779.53042
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.
MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds