On Symmetric Brackets Induced by Linear Connections

• Main Author: Bogdan Balcerzak
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Symmetry
• Subjects: skew-symmetric algebroid; almost Lie algebroid; anchored vector bundle; connection; symmetric product; symmetrized covariant derivative
• Online Access: https://www.mdpi.com/2073-8994/13/6/1003

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.