Canonical connections attached to generalized quaternionic and para-quaternionic structures

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 $TM\oplus T^{\ast }M$ 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 ${\hat{\nabla }}^{\ast }$-parallel, where $\hat{\nabla }$ is an affine connection on $TM\oplus T^{\ast }M$ induced by $\nabla$, and ${\hat{\nabla }}^{\ast }$ is its generalized dual connection with respect to a bilinear form $\check{h}$ on $TM\oplus T^{\ast }M$ 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 ${\hat{\nabla }}^{\ast }$. Finally we prove that the generalized affine connection that parallelizes certain families of generalized almost complex and almost product structures is preserved.

37 pages

References in corpus (1)

• $\alpha$-connections in generalized geometry