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In this review paper, we recall, in a unifying manner, our recent results concerning the Lie symmetries of nonlinear Fokker-Plank equations, associated to the (weighted) Tsallis and Kaniadakis entropies. The special values of the Tsallis parameters, highlighted by the classification of these symmetries, clearly indicate algebraic and geometric invariants which differentiate the Lie algebras involved. We compare these values with the ones previously obtained by several authors, and we try to establish connections between our theoretical families of entropies and specific entropies arising in several applications found in the literature.

We look for new geometric invariants associated, in canonical ways, to vector fields on differentiable manifolds, as special sets of affine connections and (semi)-Riemannian metrics. This approach is motivated by the need of geometrization of autonomous ODEs systems (and, to some extent, of the non-autonomous ones). [ABSTRACT FROM AUTHOR]

We investigate the elliptic curves, expressed in the Weierstrass form, from a differential geometric viewpoint. The curvature function is carefully studied and bounds for the number of vertices of the elliptic curves are given. As a generalization, a program for the classification of algebraic curves is sketched, using geometric invariants. [ABSTRACT FROM AUTHOR]

In [1] we denned the notion of Pareto manifolds, as a tool for generalizing classical methods of multi-criteria optimization to the framework of modern Differential geometry. The present paper contains more details and examples of Pareto manifolds, and of Pareto maxima on manifolds. [ABSTRACT FROM AUTHOR]

Following [12] and [14], the Lie groups may be classified into four families, with respect to the property of the invariant vector fields to be geodesic. A detailed study in dimensions 2 and 3 ([13]) led to many examples of particular invariant vector fields ξ geometrized, in this manner, by means of some properly chosen invariant Riemannian metrics g. In this paper we extend the previous investigation, and analyse the behavior of all the invariant vector fields ξ and of all their associated invariant Riemannian metrics g. The 2-dimensional case is completely solved: given a left invariant field of forces ξ on a 2-dimensional Lie group G, we determine all the metrics g 2 Riem(G; ξ) such that ξ becomes geodesic in (G; g) and all the possible trajectories of free fall particles moving in the "Universe" (G; g). [ABSTRACT FROM AUTHOR]