Abstract
In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.
Highlights
The Exterior Derivative Operator and the Symmetrized Covariant DerivativeWe introduce the concepts of a symmetric bracket and the related mapping ds and the symmetric Lie derivative defined on the whole tensor bundle of a given skew-symmetric algebroid
Introduction by Linear ConnectionsSymmetryLet M be a differential manifold and Sk T ∗ M denote the k-th symmetric power L k ∗of the cotangent bundle of M
We show that the condition for connections with totally skew-symmetric torsion to be compatible with the metric is that the Lie derivative of the metric should be equal to the minus of the symmetric Lie derivative of the metric
Summary
We introduce the concepts of a symmetric bracket and the related mapping ds and the symmetric Lie derivative defined on the whole tensor bundle of a given skew-symmetric algebroid. Let ( A, $ A , [·, ·]) be a skew-symmetric algebroid over a manifold M equipped with a pseudo-Riemannian metric g ∈ Γ(S2 A∗ ) in the vector bundle A and an A-connection ∇. The formula in Theorem 2 gives an explicit one of symmetric bracket defined by any metric connection with totally skew-symmetric torsion. Let ∇ be any metric A-connection in A with totally skew-symmetric torsion with respect to a pseudo-Riemannian metric g.
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