Twistors and Cartan connections
The main result of the paper gives a general approach for constructing a twistor space with a CR-structure for a manifold with a geometric structure. It states the following: Let \(G\) be a closed subgroup of a Lie group \(L\) and let \(W\) be a closed subgroup of the complexification \(L^{{\mathbb C}}\). Set \(K=W\cap G\). Let \(\pi:P\to M\) be a principal \(G\)-bundle over a manifold \(M\) and \({ k}\) a Cartan connection of type \(L\) on \(P\). If \(K=W\cap L\), then there exists a natural CR-structure on the manifold \(Z=P/K\). This CR-structure is integrable if and only if \(\Omega({ k}^{-1}({m}), { k}^{-1}({ m}))\subset { m}\) where \(\Omega\) is the curvature form of the Cartan connection \({ k}\) and \({ m}=\operatorname {Lie} W\) is the Lie algebra of \(W\). The integrability condition is further discussed in the case when the Lie algebra \(\operatorname {Lie} L\) is graded and \(\operatorname {Lie} G\), \(\operatorname {Lie} W\) are graded subalgebras. Finally, the authors present two applications of their construction -- for manifolds with a \(G\)-structure of the so-called CR-type and for the manifold of the infinite \(1\)-coframes of a given manifold.
scientific article; zbMATH DE number 1530054