Almost CR quaternionic manifolds and their immersibility in HP^n

TL;DR

This paper establishes criteria for when almost CR quaternionic manifolds can be locally immersed into quaternionic projective space, and constructs a deformation that cannot be immersed, advancing understanding of quaternionic CR geometry.

Contribution

It applies integrability conditions to quaternionic CR geometry, providing new necessary and sufficient criteria for immersibility and constructing a non-immersible deformation.

Findings

Derived criteria for local immersions into HP^n.

Constructed a deformation of quaternionic contact structure without immersibility.

Enhanced understanding of the geometry of quaternionic CR manifolds.

Abstract

We apply the general theory of codimension one integrability conditions for $G$-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.