Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

In this paper, we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.

     

Isometries, Geodesics and Jacobi Fields of Lorentzian Heisenberg Group

The paper is mainly devoted to determine the groups of isometries of the Heisenberg group endowed with each of the three left invariant Lorentzian metrics which are possible on it; also, an explicit computation of all the isometries for the (two) non flat Lorentzian metrics is done. Moreover, explicit formulas for the geodesic curves and the Jacobi vector fields for each of these three Lorentzian metrics are computed.     

Decomposing Ends of Locally Finite Graphs

An important invariant of translations of infinite locally finite graphs is that of a direction as introduced by Halin . This invariant gives not much information if the translation is not a proper one. A new refined concept of directions is investigated. A double ray D of a graph X is said to be metric, if the distance metrics in D and X on V(D) are equivalent. It is called geodesic, if these metrics are equal. The translations leaving some metric double ray invariant are characterized. Using a result of Polat and Watkins , we characterize the translations leaving some geodesic double ray invariant.     

On geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1

The present paper is devoted to the problem of (local) geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on generic corank 1 distributions. Using the Pontryagin maximum principle, we treat Riemannian and sub-Riemannian cases in a unified way and obtain some algebraic necessary conditions for the geodesic equivalence of (sub-)Riemannian metrics. In this way, first we obtain a new elementary proof of the classical Levi-Civita theorem on the classification of all Riemannian geodesically equivalent metrics in a neighborhood of the so-called regular (stable) point w.r.t. these metrics. Second, we prove that sub-Riemannian metrics on contact distributions are geodesically equivalent iff they are constantly proportional. Then we describe all geodesically equivalent sub-Riemannian metrics on quasi-contact distributions. Finally, we give a classification of all pairs of geodesically equivalent Riemannian metrics on a surface that are proportional at an isolated point. This is the simplest case, which was not covered by Levi-Civita’s theorem. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.     

Euler equations on homogeneous spaces and Virasoro orbits

We show that the following three systems related to various hydrodynamical approximations: the Korteweg-de Vries equation, the Camassa-Holm equation, and the Hunter-Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces.We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.     

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.     

Global structure and geodesics for Koenigs superintegrable systems

We present a new derivation of the local structure of Koenigs metrics using a framework laid down by Matveev and Shevchishin. All of these dynamical systems allow for a potential preserving their superintegrability (SI) and most of them are shown to be globally defined on either ?² or ?². Their geodesic flows are easily determined thanks to their quadratic integrals. Using Carter (or minimal) quantization, we show that the formal SI is preserved at the quantum level and for two metrics, for which all of the geodesics are closed, it is even possible to compute the classical action variables and the point spectrum of the quantum Hamiltonian.     

Curvature of invariant metrics on Pyatetskii-Shapiro's generalized homogeneous domains

We study the curvature of invariant metrics on the generalization of the classical homogeneous domain of Pyatetskii-Shapiro, as given by D'Atri in [3]. We obtain all invariant Kähler metrics of either, nonpositive sectional curvature or nonpositive holomorphic sectional curvature, and determine the corresponding connected groups of isometries in each case. This yields a continuous family of nonsymmetric homogeneous Kähler metrics with nonpositive curvature.Supported in part by CONICOR and SECyT (UNC).     

Flat metrics are strict local minimizers for the polynomial entropy

As we have proved in [11], the geodesic flows associated with the flat metrics on $ \mathbb{T}^2 $ minimize the polynomial entropy hpol. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated with flat metrics are local strict minima for hpol. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.     

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

In this paper we show that the geodesic flow on a compact locally symmetric space of nonpositive curvature has a unique invariant measure of maximal entropy. As an application to dynamics we show that closed geodesics are uniformly distributed with respect to this measure. Furthermore, we prove that the volume entropy is minimized at a compact locally symmetric space of nonpositive curvature among all conformally equivalent metrics with the same total volume.     

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).     

On the existence of equivalent finite invariant measures

Summary Necessary and sufficient conditions are given for the existence of a finite measure which is equivalent to a given measure and invariant with respect to each transformation in a given commutative semigroup of measurable null-invariant point transformations. This result was already known for denumerably generated semigroups. A complementary result is proved which states that if one such equivalent measure exists, then there exists a unique equivalent measure which agrees with the original measure on the invariant sets.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.     

Integration of Geodesic Flows on Homogeneous Spaces: The Case of a Wild Lie Group

We obtain necessary and sufficient conditions for the integrability in quadratures of geodesic flows on homogeneous spaces M with invariant and central metrics. The proposed integration algorithm consists in using a special canonical transformation in the space T ^* M based on constructing the canonical coordinates on the orbits of the coadjoint representation and on the simplectic sheets of the Poisson algebra of invariant functions. This algorithm is applicable to integrating geodesic flows on homogeneous spaces of a wild Lie group.     

Some Remarks on Quantum Limits on Zoll Manifolds

We study the concentration of eigenfunctions of the Laplace–Beltrami operator on manifolds all whose geodesics are closed (the so-called Zoll manifolds). Some results on the structure of the set of invariant semiclassical measures associated to sequences of eigenfunctions are given. Among these, we show that any probability measure on the unit tangent bundle of a compact rank-one symmetric space that is invariant by the geodesic flow may be realized as the semiclassical measure of a sequence of eigenfunctions of the Laplacian. This extends a previous result of Jakobson and Zelditch on spheres.     

A classification of invariant distributions and convergence of imprecise Markov chains

We analyse the structure of imprecise Markov chains and study their convergence by means of accessibility relations. We first identify the sets of states, so-called minimal permanent classes, that are the minimal sets capable of containing and preserving the whole probability mass of the chain. These classes generalise the essential classes known from the classical theory. We then define a class of extremal imprecise invariant distributions and show that they are uniquely determined by the values of the upper probability on minimal permanent classes. Moreover, we give conditions for unique convergence to these extremal invariant distributions.     

Geometric groups. I

We define a geometry on a group to be an abelian semigroup of symmetric open sets with certain properties. Examples include well-known structures such as invariant Riemannian metrics on Lie groups, hyperbolic groups, and valuations on fields. In this paper we are mostly concerned with geometries where the semigroup is isomorphic to the positive reals, which for Lie groups come from invariant Finsler metrics. We explore various aspects of these geometric groups, including a theory of covering groups for arcwise connected groups, algebraic expressions for invariant metrics and inner metrics, construction of geometries with curvature bounded below, and finding geometrically significant curves in path homotopy classes.

     

On the geometry of generalized Gaussian distributions

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q=1 case these are the normal distributions. Some methods to endow this parameter space with a Riemannian metric is presented: the second derivative of the q-Rényi entropy, the Tsallis entropy, and the relative entropy give rise to a Riemannian metric, the Fisher information matrix is a natural Riemannian metric, and there are some geometrically motivated metrics which were studied by Siegel, Calvo and Oller, Lovri?, Min-Oo and Ruh. These metrics are different; therefore, our differential geometrical calculations are based on a new metric with parameters, which covers all the above-mentioned metrics for special values of the parameters, among others. We also compute the geometrical properties of this metric, the equation of the geodesic line with some special solutions, the Riemann and Ricci curvature tensors, and the scalar curvature. Using the correspondence between the volume of the geodesic ball and the scalar curvature we show how the parameter q modulates the statistical distinguishability of close points. We show that some frequently used metrics in quantum information geometry can be easily recovered from classical metrics.     

Generalized-periodic motions of nonautonomous systems

The definition and existence criterion are given for the generalized-periodic motions of a certain wide class of systems. The class contains all the systems that can be characterized by the classical periodic operator of displacement, the systems generated by the Volterra integral equations, and some others. A relationship is established between generalized-periodic motions and integral invariant sets.