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Register a new userOn homogeneous and symmetric CR manifolds
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We consider canonical fibrations and algebraic geometric structures on homogeneous CR manifolds, in connection with the notion of CR algebra. We give applications to the classifications of left invariant CR structures on semisimple Lie groups and of CR-symmetric structures on complete flag varieties.
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We consider a class of compact homogeneous CR manifolds, that we call $mathfrak n$-reductive, which includes the orbits of minimal dimension of a compact Lie group $K_0$ in an algebraic homogeneous variety of its complexification $K$. For these manifolds we define canonical equivariant fibrations onto complex flag manifolds. The simplest example is the Hopf fibration $S^3tomathbb{CP}^1$. In general these fibrations are not $CR$ submersions, however they satisfy a weaker condition that we introduce here, namely they are CR-deployments.
Using a bigraded differential complex depending on the CR and pseudohermitian structure, we give a characterization of three-dimensional strongly pseudoconvex pseudo-hermitian CR-manifolds isometrically immersed in Euclidean space $mathbb{R}^n$ in terms of an integral representation of Weierstrass type. Restricting to the case of immersions in $mathbb{R}^4$, we study harmonicity conditions for such immersions and give a complete classification of CR-pluriharmonic immersions.
We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M. We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space SL(3, C)/B, where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern-Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns-Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.
In this paper, we give some rigidity results for both harmonic and pseudoharmonic maps from CR manifolds into Riemannian manifolds or Ku007fahler manifolds. Some basicity, pluriharmonicity and Siu-Sampson type results are established for both harmonic maps and pseudoharmonic maps.
We construct $Q$-curvature operators on $d$-closed $(1,1)$-forms and on $overline{partial}_b$-closed $(0,1)$-forms on five-dimensional pseudohermitian manifolds. These closely related operators give rise to a new formula for the scalar $Q$-curvature. As applications, we give a cohomological characterization of CR five-manifolds which admit a $Q$-flat contact form; and we show that every closed, strictly pseudoconvex CR five-manifold with trivial first real Chern class admits a $Q$-flat contact form provided the $Q$-curvature operator on $overline{partial}_b$-closed $(0,1)$-forms is nonnegative.
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