Canonical connections attached to generalized quaternionic and para-quaternionic structures

TL;DR

This paper introduces canonical connections for generalized quaternionic and para-quaternionic structures, characterizes their integrability, and explores conditions for their parallelism with respect to induced affine connections.

Contribution

It establishes the existence of a canonical, torsion-free connection that parallelizes these structures and links them to generalized Obata and dual connections, advancing the understanding of their geometric properties.

Findings

Existence of a canonical torsion-free connection for generalized quaternionic structures.

Characterization of integrability conditions for these structures.

Identification of the canonical connection as the generalized Obata connection in quaternionic cases.

Abstract

We put into light some generalized almost quaternionic and almost para-quaternionic structures and characterize their integrability with respect to a $\nabla$-bracket on the generalized tangent bundle $TM\oplus T^*M$ of a smooth manifold $M$, defined by an affine connection $\nabla$ on $M$. Also, we provide necessary and sufficient conditions for these structures to be $\hat \nabla$-parallel and $\hat \nabla^*$-parallel, where $\hat \nabla$ is an affine connection on $TM\oplus T^*M$ induced by $\nabla$, and $\hat\nabla^*$ is its generalized dual connection with respect to a bilinear form $\check h$ on $TM\oplus T^*M$ induced by a non-degenerate symmetric or skew-symmetric $(0,2)$-tensor field $h$ on $M$. As main results, we establish the existence of a canonical connection associated to a generalized quaternionic and to a generalized para-quaternionic structure, i.e., a torsion-free generalized affine connection that parallelizes these structures. We show that, in the quaternionic case, the canonical connection is the generalized Obata connection and that on a quasi-statistical manifold $(M,h,\nabla)$, an integrable $h$-symmetric and $\nabla$-parallel $(1,1)$-tensor field gives rise to a generalized para-quaternionic structure whose canonical connection is precisely $\hat \nabla^*$. Finally we prove that the generalized affine connection that parallelizes certain families of generalized almost complex and almost product structures is preserved.