Almost Hermitian structures on the tangent bundle of almost symplectic manifold

Abstract

where Γij are the connection coefficients of ∇. Taking the cyclic permutation of the indices in (1) and summing up the resulting equations, we obtain ωpkS p ij + ωpjS p ki + ωpiS p jk = ∂iωjk + ∂jωki + ∂kωij, where Sk ij = Γ k ij − Γji are the components of the torsion tensor S of the connection ∇. Hence follows that, if an almost symplectic structure is not a symplectic structure (dω = 0), then the connection necessarily has torsion. For a given symplectic structure ω, infinitely many connections compatible with ω exist. The coefficients of these connections can be written as follows [1]: