On Symmetric Brackets Induced by Linear Connections

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian structure, describe the induced by connections totally torsion in language of Lie derivatives and differentials functions. We formulate generalization fundamental theorem Riemannian geometry. particular, obtain an explicit formula Levi-Civita connection. also present some almost Hermitian manifolds first canonical using brackets. define bracket smooth functions structure show that it has properties analogous to Hamiltonian vector fields manifolds.