2025/06/29 by Abdelhak Abouqateb, Abouqateb, Abdelhak, Othmane Dani +1
Mathematics · Physics and Astronomy · #14M17 #53B05 #53C30 #53D05 #70G45 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2506.23211
openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A symplectic reductive homogeneous space is a pair (G/H,Ω), where G/H is a reductive homogeneous G-space and Ω is a G-invariant symplectic form on it. The main examples include symplectic Lie groups, symplectic symmetric spaces, and flag manifolds. This paper focuses on the existence of a natural symplectic connection on (G/H,Ω). First, we introduce a family \∇a,b\(a,b)∈ℝ2 of G-invariant connection on G/H, and establish that ∇0,1 is flat if and only if (G/H,Ω) is locally a symplectic Lie group. Next, we show that among all \∇a,b\(a,b)∈ℝ2, there exists a unique symplectic connection, denoted by ∇s, corresponding to a=b=\tfrac13, a fact that seems to have previously gone unnoticed. We then compute its curvature and Ricci curvature tensors. Finally, we demonstrate that the SU(3)-invariant preferred symplectic connection of the Wallach flag manifold SU(3)/\mathbbT2 (from Cahen-Gutt-Rawnsley) coincides with the natural symplectic connection ∇s, which is furthermore Ricci-parallel.