Generalizations of differential geometric structures on pre-Leibniz algebroids

Algebroids are mathematical structures that generalize the tangent bundle of a manifold. Hence, they can be considered as an appropriate mathematical framework for generalizations of differential geometric structures on a manifold. For example, metric-affine geometry can be constructed on a Lie algebroid of which the tangent bundle constitutes an example. On the other hand, generalized geometry can be written on a Courant algebroid. As differential geometric structures play a fundamental role in classical field theories, these algebroids are also crucial for physical purposes. For instance, general relativity can be expressed in terms of metric-affine geometric structures, whereas generalized geometry is closely related to the double field theory formulation of bosonic string theory. There are other algebroid structures such as metric algebroids, higher-Courant algebroids that are widely used in string and M theories. It is then natural to seek for a comprehensive structure that includes all of these algebroids as special cases. Local pre-Leibniz algebroids, and in particular anti-commutable pre-Leibniz algebroids, are such general ones. Differential geometry on these algebroids is of great interest, as they create a chance to work with all of these individual structures at once. In this thesis, various differential geometric structures on pre-Leibniz algebroids are studied. At the fundamental level, metric-connection geometries are constructed, and their properties are examined. On anti-commutable pre-Leibniz algebroids, for a class of connections that we call admissible, many relations that hold in the usual differential geometry are proven. These include Bianchi and Ricci identities, Cartan structure equations and the decomposition of a connection in terms of its torsion and non-metricity. Moreover, we introduce statistical, Hessian, conformal, projective and Weyl structures on pre-Leibniz algebroids, and prove some results analogous to the manifold setting.