2025/11/18 by Borówka, Aleksandra, Chrysikos, Ioannis
Mathematics · Physics and Astronomy · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Advanced Differential Geometry Research
paper · doi:10.48550/arxiv.2511.14392
We show that a metric f-manifold (M2n+s, ϕ, ξi, ηj, g) satisfying the property [ξi, ξj]=0 for all i, j∈\1, …, s\ admits a metric connection ∇ with skew-torsion T preserving the structure if and only if each Reeb vector field ξi is Killing and the Nijenhuis tensor N(1) is totally skew-symmetric. The connection is then uniquely determined and its torsion 3-form T is given by T=∑i=1sηi\wedge\rm dηi+\rm dϕF+N(1)-∑i=1s(ηi\wedge(ξi\lrcorner N(1))) , where \rm dϕF:=-\rm d F∘ϕ. This provides a natural higher-dimensional generalization of the adapted connections with skew-torsion on almost Hermitian manifolds (case s=0) and almost contact metric manifolds (case s=1) presented in [FrIv]. We further prove that a contact metric f-manifold (M2n+s, ϕ, ξi, ηj, g), also known as an almost S-manifold, admits such a connection if and only if M2n+s is an S-manifold, that is, a normal contact metric f-manifold. In this case we show that the torsion 3-form T, which is given by T=∑i=1sηi\wedge\rm dηi, is ∇-parallel. Thus, for s≥ 2, we construct a broad new class of geometries with parallel skew-torsion in all dimensions ≥ 4, both even and odd. These geometries differ from the Sasakian case (s=1) also by the fact that their torsion 3-form T is degenerate. We finally describe examples with s=2, s=3 and s=4, relying on the Lie groups U(2) and U(3), and a construction of S-manifolds presented in [DL05]. For the latter case and the case of U(2) we compute the holonomy algebra of the connection ∇ and show that ∇ is an Ambrose-Singer connection, that is, ∇ T=0=∇ R∇.