About the geometry of almost para-quaternionic manifolds

TL;DR

This paper establishes criteria for the integrability of almost para-quaternionic structures on manifolds, linking it to twistor and reflector space structures, and demonstrates the abundance of compatible complex and para-complex structures.

Contribution

It provides new integrability criteria for almost para-quaternionic manifolds and connects these to twistor and reflector space structures, advancing understanding of their geometric properties.

Findings

Criteria for integrability in terms of sections of P

Relation between integrability and twistor/reflector spaces

Existence of compatible complex and para-complex structures

Abstract

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is para-quaternionic.