Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

On Birkhoff Theorems for Lagrangian invariant tori with closed orbits

We show that a C¹ torus that is homologous to the zero section, invariant by the geodesic flow of a symmetric Finsler metric in T², and possesses closed orbits is a graph of the canonical projection. This result, together with the result obtained by Bialy in 1989 for continuous invariant tori without closed orbits of symmetric Finsler metrics in T², shows that the second Birkhoff Theorem holds for C¹ Lagrangian invariant tori of symmetric Finsler metrics in the two torus. We also study the first Birkhoff Theorem for continuous invariant tori of Finsler metrics in T² and give some sufficient conditions for a continuous minimizing torus with closed orbits to be a graph of the canonical projection. Partially supported by CNPq, FAPERJ, TWAS     

Ricci soliton homogeneous nilmanifolds

We study a notion weakening the Einstein condition on a left invariant Riemannian metric g on a nilpotent Lie groupN. We consider those metrics satisfying Ric for some and some derivationD of the Lie algebra ofN, where Ric denotes the Ricci operator of . This condition is equivalent to the metric g to be a Ricci soliton. We prove that a Ricci soliton left invariant metric on N is unique up to isometry and scaling. The following characterization is also given: (N,g) is a Ricci soliton if and only if (N,g) admits a metric standard solvable extension whose corresponding standard solvmanifold is Einstein. This gives several families of new examples of Ricci solitons. By a variational approach, we furthermore show that the Ricci soliton homogeneous nilmanifolds (N,g) are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set. Received August 24, 1999 / Revised October 2, 2000 / Published online February 5, 2001     

Local monotonicity of Riemannian and Finsler volume with respect to boundary distances

We show that the volume of a simple Riemannian metric on D ⁿ is locally monotone with respect to its boundary distance function. Namely if g is a simple metric on D ⁿ and g′ is sufficiently close to g and induces boundary distances greater or equal to those of g, then vol(D ⁿ , g′) ≥ vol(D ⁿ , g). Furthermore, the same holds for Finsler metrics and the Holmes–Thompson definition of volume. As an application, we give a new proof of injectivity of the geodesic ray transform for a simple Finsler metric.     

Embeddings of Gromov hyperbolic spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. Submitted: October 1998, Revised version: January 1999.     

Integrability Criterion of Geodesical Equivalence. Hierarchies

We prove that the Riemannian metrics g and (given in `general position") are geodesically equivalent if and only if some canonically given functions are pairwise commuting integrals of the geodesic flow of the metric g. This theorem is a multidimensional generalization of the well-known Dini theorem proved in the two-dimensional case. A hierarchy of completely integrable Riemannian metrics is assigned to any pair of geodesically equivalent Riemannian metrics. We show that the metrics of the standard ellipsoid and the Poisson sphere lie in such an hierarchy.     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

Einstein metrics and complex singularities

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkähler gravitational instantons, but we focus on a different class of singularities. We show that any resolution X of an isolated cyclic quotient singularity admits a complete scalar-flat Kähler metric (which is hyperkähler if and only if K_X is trivial), and that if K_X is strictly nef, then X also admits a complete (non-Kähler) self-dual Einstein metric of negative scalar curvature. In particular, complete self-dual Einstein metrics are constructed on simply-connected non-compact 4-manifolds with arbitrary second Betti number.Deformations of these self-dual Einstein metrics are also constructed: they come in families parameterized, roughly speaking, by free functions of one real variable.All the metrics constructed here are toric (that is, the isometry group contains a 2-torus) and are essentially explicit. The key to the construction is the remarkable fact that toric self-dual Einstein metrics are given quite generally in terms of linear partial differential equations on the hyperbolic plane.     

Isometry Groups of Proper Hyperbolic Spaces

Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not elementary then for every p ∈ (1, ∞) the second continuous bounded cohomology group H²_cb(G, L^p(G)) does not vanish. As an application, we derive some structure results for closed subgroups of Iso(X). Partially supported by Sonderforschungsbereich 611.     

Characterizing hyperbolic spaces and real trees

Let X be a geodesic metric space. Gromov proved that there exists ε ₀ > 0 such that if every sufficiently large triangle Δ satisfies the Rips condition with constant ε ₀ · pr(Δ), where pr(Δ) is the perimeter of Δ, then X is hyperbolic. We give an elementary proof of this fact, also giving an estimate for ε ₀. We also show that if all the triangles D í X{\Delta \subseteq X} satisfy the Rips condition with constant ε ₀ · pr(Δ), then X is a real tree. Moreover, we point out how this characterization of hyperbolicity can be used to improve a result by Bonk, and to provide an easy proof of the (well-known) fact that X is hyperbolic if and only if every asymptotic cone of X is a real tree.     

Products of Hyperbolic Metric Spaces

Let (X _i d _i ), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a 'hyperbolic product' X ₁× _h X ₂ which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.     

Invariant Metrics on G-Spaces

Let X be a G-space such that the orbit space X/G is metrizable. Suppose a family of slices is given at each point of X. We study a construction which associates, under some conditions on the family of slices, with any metric on X/G an invariant metric on X. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.     

Path of quasi-means as a geodesic

As a generalization of the Hiai-Petz geometries, we discuss two types of them where the geodesics are the quasi-arithmetic means and the quasi-geometric means respectively. Each derivative of such a geodesic might determine a new relative operator entropy. Also in these cases, the Finsler metric can be induced by each unitarily invariant norm. If the norm is strictly convex, then the geodesic is the shortest. We also give examples of the shortest paths which are not the geodesics when the Finsler metrics are induced by the Ky Fan k-norms.     

Hilbert metrics and Minkowski norms

It is shown that the Hilbert geometry (D, h_D) associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert’s metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.     

Well-posedness of modified Camassa-Holm equations

The Camassa-Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant H¹ metric. We derive the geodesic equations on that group with respect to the invariant Hk metric, which we call the modified Camassa-Holm equation, and then study the well-posedness and dynamics of a modified Camassa-Holm equation on the unit circle S, which has some significant difference from that of Camassa-Holm equation, e.g., it does not admit finite time blowup solutions.     

Spaces of Densely Continuous Forms, USCO and Minimal USCO Maps

Our paper studies the topology of uniform convergence on compact sets on the space of densely continuous forms (introduced by Hammer and McCoy (1997)), usco and minimal usco maps. We generalize and complete results from Hammer and McCoy (1997) concerning the space D(X,Y) of densely continuous forms from X to Y. Let X be a Hausdorff topological space, (Y,d) be a metric space and D _k (X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: D _k (X,Y) is metrizable iff D _k (X,Y) is first countable iff X is hemicompact. This result gives also a positive answer to question 4.1 of McCoy (1998). If moreover X is a locally compact hemicompact space and (Y,d) is a locally compact complete metric space, then D _k (X,Y) is completely metrizable, thus improving a result from McCoy (1998). We study also the question, suggested by Hammer and McCoy (1998), when two compatible metrics on Y generate the same topologies of uniform convergence on compact sets on D(X,Y). The completeness of the topology of uniform convergence on compact sets on the space of set-valued maps with closed graphs, usco and minimal usco maps is also discussed.     

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.     

A Generalization of the Hutchinson Measure

The Hutchinson measure is the invariant measure associated with an iterated function system with probabilities. Generalized iterated function systems (GIFS) are generalizations of iterated function systems which are obtained by considering contractions from X × X to X, rather than contractions from a metric space X to itself. Along the lines of this generalization we consider GIFS with probabilities. In this paper we prove the existence of an analogue of Hutchinson measure associated with a GIFS with probabilities and present some of its properties. The work was supported by CNCSIS grant 8A;1067/2006.     

A note on chaos via Furstenberg family couple

The concepts of the first type of distributional chaos in the Tan-Xiong sense (Abbrev. DC1 in the Tan-Xiong sense), the second type of strong-distributional chaos (Abbrev. strong DC2) and the third type of strong-distributional chaos (Abbrev. strong DC3) were introduced by Tan et al. [F. Tan, J. Xiong. Chaos via Furstenberg family couple, Topology Appl. (2008), doi:10.1016/j.topol.2008.08.006] for continuous maps of a metric space. However, it turns out that, for continuous maps of a compact metric space, the three mutually nonequivalent versions of distributional chaos can be discussed. Let X be a compact metric space and f:X→X a continuous map. In this paper, we show that for any integer N>0, f is strong DC2 (resp. strong DC3) if and only if f^N is strong DC2 (resp. strong DC3). We also show that the above three versions of distributional chaos are topological conjugacy invariant. In addition, as an application, we present an example.     

Geodesic symmetries of homogeneous Kähler Manifolds

D'Atri and Nickerson [6], [7] have given necessary conditions for the geodesic symmetries of a Riemannian manifold to preserve the volume element. We use their results to show that ifG is a compact simple Lie group,T is a maximal torus ofG, andG/T is not symmetric, then anyG-invariant Kähler metric onG/T does not have volume-preserving geodesic symmetries. From the Kähler/de Rham decomposition of a compact homogeneous Kähler manifold [8], our result extends to the invariant Kähler metrics on a quotient of a compact connected Lie group by a maximal torus. In proving these results we compute directly the Ricci tensor of anyG-invariant Kähler metric onG/T forG compact connected andT a maximal torus ofG. The result is an explicit formula giving the value of the Ricci tensor elements in terms of the root structure of the Lie algebra ofG.