Geometric Structures Generated by the Same Dynamics. Recent Results and Challenges

Abstract

The main contribution of this review is to show some relevant relationships between three geometric structures on a connected Lie group G, generated by the same dynamics. Namely, Linear Control Systems, Almost Riemannian Structures, and Degenerate Dynamical Systems. These notions are generated by two ordinary differential equations on G: linear and invariant vector fields. A linear vector field on G is determined by its flow, a 1-parameter group of Aut(G), the Lie group of G-automorphisms. An invariant vector field is just an element of the Lie algebra g of G. The Jouan Equivalence Theorem and the Pontryagin Maximum Principal are instrumental in this setup, allowing the extension of results from Lie groups to arbitrary manifolds for the same kind of structures which satisfy the Lie algebra finitude condition. For each structure, we present the first given examples; these examples generate the systems in the plane. Next, we introduce a general definition for these geometric structures on Euclidean spaces and G. We describe recent results of the theory. As an additional contribution, we conclude by formulating a list of open problems and challenges on these geometric structures. Since the involved dynamic comes from algebraic structures on Lie groups, symmetries are present throughout the paper.