Quaternionic-like manifolds and homogeneous twistor spaces

TL;DR

This paper introduces quaternionic-like manifolds, a new class generalizing quaternionic geometry, and explores their properties, including the existence of unique heaven spaces and their relation to homogeneous complex manifolds with embedded spheres.

Contribution

It defines quaternionic-like manifolds, studies their structure, and establishes the existence of unique heaven spaces, expanding the understanding of quaternionic geometry and homogeneous complex manifolds.

Findings

Existence of unique heaven spaces for quaternionic-like manifolds.

Construction of homogeneous complex manifolds with embedded spheres.

Quaternionic-like manifolds encompass CR quaternionic and ρ-quaternionic manifolds.

Abstract

Motivated by the quaternionic geometry corresponding to the homogeneous complex manifolds endowed with (holomorphically) embedded spheres, we introduce and initiate the study of the `quaternionic-like manifolds'. These contain, as particular subclasses, the CR quaternionic and the $\rho$-quaternionic manifolds. Moreover, the notion of `heaven space' finds its adequate level of generality in this setting: (essentially) any real analytic quaternionic-like manifold admits a (germ) unique heaven space, which is a $\rho$-quaternionic manifold. We, also, give a natural construction of homogeneous complex manifolds endowed with embedded spheres, thus, emphasizing the abundance of the quaternionic-like manifolds.