2023/02/10 by Adara M. Blaga, Blaga, Adara M., Antonella Nannicini +1
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2302.05239
openalex publication_date 2023/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We put into light some generalized almost quaternionic and almost para-quaternionic structures and characterize their integrability with respect to a ∇-bracket on the generalized tangent bundle TM⊕ T^*M of a smooth manifold M, defined by an affine connection ∇ on M. Also, we provide necessary and sufficient conditions for these structures to be ∇-parallel and ∇^*-parallel, where ∇ is an affine connection on TM⊕ T^*M induced by ∇, and ∇^* is its generalized dual connection with respect to a bilinear form \check h on TM⊕ 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,∇), an integrable h-symmetric and ∇-parallel (1,1)-tensor field gives rise to a generalized para-quaternionic structure whose canonical connection is precisely ∇^*. Finally we prove that the generalized affine connection that parallelizes certain families of generalized almost complex and almost product structures is preserved.