'Anti-Commutable' Pre-Leibniz Algebroids and Admissible Connections

Abstract

The concept of algebroid is convenient for constructions of geometrical frameworks. For example, metric-affine and generalized geometries can be written on Lie and Courant algebroids, respectively. Furthermore, string theories might make use of many other algebroids such as metric algebroids, higher-Courant algebroids, or conformal Courant algebroids. Working on the possibly most general algebroid structure is fruitful as it creates a chance to study all of them at once. Local pre-Leibniz algebroids are such general ones in which metric-connection geometries are possible to construct. Yet, the existence of the locality operator necessitates the modification of torsion and curvature operators to achieve tensorial quantities. In this paper, this modification is explained from the point of view that the modification is applied to the bracket instead. This leads one to consider `anti-commutable' pre-Leibniz algebroids whose bracket satisfies a property defined with respect to a choice of an equivalence class of connections. These `admissible' connections are claimed to be the necessary ones for a geometry on algebroids because one can prove many desirable properties for them. For instance, we prove the first and second Bianchi identities, Ricci identity, Cartan structure equations, the construction of Levi-Civita connections, the decomposition of connection in terms of torsion and non-metricity. These all are possible because the modified bracket becomes anti-symmetric for an admissible connection so that one can apply the machinery of almost- or pre-Lie algebroids. We investigate various algebroid structures from the literature and show that they admit admissible connections which are metric-compatible in some generalized sense. Moreover, we prove that local pre-Leibniz algebroids that are not anti-commutable cannot be equipped with a torsion-free, and in particular Levi-Civita, connection.