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The metric-affine and generalized geometries, respectively, are arguably theappropriate mathematical frameworks for Einstein's theory of gravity and thelow-energy effective massless oriented closed bosonic string field theory. Infact, mathematical structures in a metric-affine geometry are written on thetangent bundle, which is itself a Lie algebroid; whereas those in generalizedgeometries introduced as the basis of double field theories, are written onCourant algebroids. The Lie, Courant and the higher Courant algebroids used inexceptional field theories, are all special cases of pre-Leibniz algebroids.Provided with some additional ingredients, the construction of such geometriescan all be carried over to regular pre-Leibniz algebroids. We define below thenotions of locality structures and locality projectors, which are some suchnecessary ingredients. In terms of these structures, $E$-metric-connectiongeometries are constructed with (possibly) a minimum number of assumptions.Certain small gaps in the literature are also filled as we go along. $E$-Koszulconnections, as a generalization of Levi-Civita connections, are going to bedefined and shown to be helpful for some results including a simplegeneralization of the fundamental theorem of Riemannian geometry. We also showthat metric-affine geometries can be constructed in a unique way as specialcases of $E$-metric-connection geometries. Moreover, generalized geometries areshown to follow as special cases, and various properties of lineargeneralized-connections are proven in the present framework. Similarly,uniqueness of the locality projector in the case of exact Courant algebroids isproven; a result that explains why the curvature operator, defined with aprojector in the double field theory literature is a necessity.

Metric-Connection Geometries on Pre-Leibniz Algebroids: A Search for Geometrical Structure in String Models