On CR singular CR images

We say that a CR singular submanifold [Formula: see text] has a removable CR singularity if the CR structure at the CR points of [Formula: see text] extends through the singularity as an abstract CR structure on [Formula: see text]. We study such real-analytic submanifolds, in which case removability is equivalent to [Formula: see text] being the image of a generic real-analytic submanifold [Formula: see text] under a holomorphic map that is a diffeomorphism of [Formula: see text] onto [Formula: see text], what we call a CR image. We study the stability of the CR singularity under perturbation, the associated quadratic invariants, and conditions for removability of a CR singularity. A lemma is also proved about perturbing away the zeros of holomorphic functions on CR submanifolds, which could be of independent interest.

Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds

In [An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23(3) (2012)], B.-Y. Chen introduced the CR δ-invariant for CR-submanifolds. Then, in [Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ-invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR-submanifolds in almost contact metric manifolds.

Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.