Statistical geometry and Hessian structures on pre-Leibniz algebroids

We introduce statistical, conjugate connection and Hessian structures on anti-commutable pre-Leibniz algebroids. Anti-commutable algebroids are special cases of local algebroids, which still general enough to include many physically motivated such as Lie, Courant, metric higher-Courant They create a natural framework for generalizations differential geometric smooth manifold. The symmetrization the bracket an algebroid satisfies certain property depending choice equivalence class connections called admissible. These admissible shown be necessary generalize aforementioned Consequently, we prove that, provided conditions, statistical equivalent when defined connections. Moreover, also show that `projected-torsion-free' connections, one can metrics structures. any structure yields structure, where these results completely parallel ones in manifold setting. mild generalization fundamental theorem geometry. $\alpha$-connections, strongly relative torsion operator, some analogous results.