CR relatives Kaehler manifolds

TL;DR

This paper explores the relationships between Kähler manifolds and CR-submanifolds, establishing conditions under which they do not share certain substructures, and demonstrating limitations on immersions into flag manifolds.

Contribution

It proves that non-sharing of Kähler submanifolds implies non-sharing of certain CR-submanifolds and shows that Levi degenerate CR-submanifolds cannot be isometrically immersed into flag manifolds.

Findings

Two Kähler manifolds without a shared Kähler submanifold do not share a Levi degenerate CR-submanifold.

Levi degenerate CR-submanifolds of mathbb{C}^n cannot be isometrically immersed into flag manifolds.

The results establish new restrictions on the geometric relationships between Kähler and CR-submanifolds.

Abstract

In this paper we show that two K\"ahler manifolds which do not share a K\"ahler submanifold, do not share either a Levi degenerate CR-submanifold with constant dimension Levi kernel. In particular, they do not share a CR-product. Further, we obtain that a Levi degenerate CR-submanifold of $\mathbb C^n$ cannot be isometrically immersed into a flag manifold.