Generalized Quaternionic Manifolds

TL;DR

This paper introduces the concept of generalized quaternionic manifolds, classifies their vector spaces, and provides examples showing their relation to complex symplectic manifolds and Einstein-Weyl spaces.

Contribution

It classifies generalized quaternionic vector spaces and demonstrates that complex symplectic and Einstein-Weyl spaces naturally admit such structures.

Findings

Any complex symplectic manifold has a natural generalized quaternionic structure.

The heaven space of a three-dimensional Einstein-Weyl space admits a generalized quaternionic structure.

The product of a complex symplectic manifold with a sphere has a natural generalized complex structure.

Abstract

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein-Weyl space. In particular, on the product $Z$ of any complex symplectic manifold $M$ and the sphere there exists a natural generalized complex structure, with respect to which $Z$ is the twistor space of $M$.