On the Lichnerowicz conjecture for CR manifolds with mixed signature

TL;DR

This paper constructs specific examples of nondegenerate CR manifolds with mixed signature that are compact or noncompact, not locally CR flat, and admit essential CR vector fields, serving as counterexamples to a CR analogue of the Lichnerowicz conjecture.

Contribution

It provides the first known counterexamples of CR manifolds with mixed signature to the Lichnerowicz conjecture, expanding understanding of CR geometry.

Findings

Existence of compact non-CR flat CR manifolds with mixed signature

Existence of noncompact non-CR flat CR manifolds with signature (1,n-1)

Counterexamples to the Lichnerowicz conjecture in CR geometry

Abstract

We construct examples of nondegenerate CR manifolds with Levi form of signature $(p,q)$, $2\leq p\leq q$, which are compact, not locally CR flat, and admit essential CR vector fields. We also construct an example of a noncompact nondegenerate CR manifold with signature $(1,n-1)$ which is not locally CR flat and admits an essential CR vector fields. These provide counterexamples to the analogue of the Lichnerowicz conjecture for CR manifolds with mixed signature.