Infinitesimmaly homogeneous manifolds with symplectic and almost contact structure groups
We study infinitesimally homogeneous affine manifolds with prescribed structure groups. First, for the real symplectic group $\operatorname{Sp}(2n,\mathbb R)$, we prove that the corresponding infinitesimally homogeneous structures are precisely flat torsion-free symplectic affine structures. Second, for \(\operatorname{U}(n)\times\{1\}\), whose \(G\)-structures are equivalent to almost contact metric data, we classify the torsion-free characteristic tensors and obtain the general torsional case by invariant deformation. The admissible torsion and inner-torsion parameters are governed by a single algebraic relation, and the horizontal distribution is contact precisely when a certain combination of these parameters does not vanish. We also describe the compatible connections and relate the torsion-free branches to trans-Sasakian geometry.