Degeneracy of Koszul Homological Series on Lie Algebroids. Production of All Affine Structures, Production of all Riemannian Foliations and Production of All Fedosov Structures

The framework of the research whose part of results are published in this work is the category of real vector bundles over finite dimensional differentiable manifolds. The objects of studies are \( \textit{gauge structures on these vector bundles} \). We are interested in dynamical properties of the holonomy groups of Koszul connections as well as on their topological properties, i.e. properties that are of homological nature. For the most part the context is the subcategory of Lie algebroids. In addition to other investigations three open problems are studied in detail. (P1-Affine Geometry): When is a Koszul connection affine connection? (P2-Riemannian Geometry): When is a Koszul connection metric connection? (P3-Fedosov Geometry): When is a Koszul connection symplectic connection? In the category of tangent Lie algebroids our homological approach leads to deep relations of our homological ingredients with the open problem of \( \textit{how to produce labeled foliations the most studied of which are Riemannian foliations} \). On a Lie algebroid we define two families of differential equations, the family of differential Hessian equations and the family of differential gauge equations. The solutions of these differential equations are implemeted to construct homological ingredients which are key tools for our studies of open problems we are concerned with. We introduce \( \textit{Koszul Homological Series}\textit{Koszul Homological Series} \). This notion is a machine for converting Obstructions whose nature is vector space into Obstructions whose nature is Homological class. We define the property of Degeneracy and the property Nondegeneracy of Koszul homological Series. The property of Degeneracy is implemented to solve problems (P1), (P2) and (P3). \( \textit{In the abundant literature on Riemannian foliatins we have only cited references directely related to the open problems which are studied using the tools which are introduced in this work. Thus the property of nondegeneracy is implemented to give a complete solution of the problem posed by E. Ghys, (P4-Differential Topology): How to produce Riemanian foliations?} \). See our Theorem 7.4 and Theorem 7.5 which are fruits of a happy conjunction between the gauge geometry and the differential topology.