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"The core of this project can be shortly (and roughly) described as project in Geometric Metric Theory and curvature equations in non-Euclidean structures. It is worthwhile from the very beginning to state clearly that, when we mention non-Euclidean structures, we refer to metric structures that are not Euclidean at any scale. Thus, the model we have in mind are not Riemannian manifolds, but bette ...