Vector fields dynamics as geodesic motion on Lie groups

We study to what extent vector fields on Lie groups may be considered as geodesic fields. For a given left invariant vector field on a Lie group, we prove there exists a Riemannian metric whose geodesics are its trajectories. When we consider left invariant metrics, differences between the Riemannian and the Lorentzian cases appear, coded by properties of the Lie algebra. To cite this article: G.T. Pripoae, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Weighted Relative Group Entropies and Associated Fisher Metrics

A large family of new α-weighted group entropy functionals is defined and associated Fisher-like metrics are considered. All these notions are well-suited semi-Riemannian tools for the geometrization of entropy-related statistical models, where they may act as sensitive controlling invariants. The main result of the paper establishes a link between such a metric and a canonical one. A sufficient condition is found, in order that the two metrics be conformal (or homothetic). In particular, we recover a recent result, established for α=1 and for non-weighted relative group entropies. Our conformality condition is “universal”, in the sense that it does not depend on the group exponential.     

Examples of invariant semi-Riemannian metrics on 4-dimensional lie groups

We consider a family of left invariant semi- Riemannian metrics on some extension of the Heisenberg group by the real line (denoted by ). We find a 3-dimensional foliation, which is minimal but not totally geodesic with respect to all the metrics of this family. Other two 3-dimensional totally geodesic (isometric) foliations on are determined. We consider also a non-holonomic 3-dimensional distribution, admitting integral surfaces which are totally geodesic in the ambiant space . Two of them are isomorphic with the two-dimensional non-commutative Lie group (which is not totally geodesic in the additive Lie groupR ⁴!). Following the different possible choices of the signatures of the metrics and the sign of the parameters, we put in evidence twelve new classes of invariant spacetime structures onR ⁴, together with their energy-momenta.