Compatible Metrics Research Articles

On a converse to Banach’s Fixed Point Theorem

We say that a metric space (X, d) possesses the Banach Fixed Point Property (BFPP) if every contraction f: X → X has a fixed point. The Banach Fixed Point Theorem states that every complete metric space has the BFPP. However, E. Behrends pointed out in 2006 that the converse implication does not hold; that is, the BFPP does not imply completeness; in particular, there is a nonclosed subset of ℝ 2 possessing the BFPP. He also asked if there is even an open example in ℝ n , and whether there is a 'nice' example in ℝ. In this note we answer the first question in the negative, the second one in the affirmative, and determine the simplest such examples in the sense of descriptive set theoretic complexity. Specifically, first we prove that if X ⊂ ℝ n is open or X ⊂ R is simultaneously F σ and G δ and X has the BFPP, then X is closed. Then we show that these results are optimal, as we give an F σ and also a G δ nonclosed example in R with the BFPP. We also show that a nonmeasurable set can have the BFPP. Our non-G δ examples provide metric spaces with the BFPP that cannot be remetrized by any compatible complete metric. All examples are in addition bounded.

The classification of nonsingular multidimensional Dubrovin-Novikov brackets

In this paper, the well-known Dubrovin-Novikov problem posed as long ago as in 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type, is solved. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. Generally speaking, such Poisson brackets are generated by nontrivial canonical special infinite-dimensional Lie algebras. In this paper, we obtain a classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differential-geometric methods. A key role in the solution of this problem is played by the theory of compatible metrics earlier constructed by the present author.

Uniqueness and Examples of Compact Toric Sasaki-Einstein Metrics

In [11] it was proved that, given a compact toric Sasaki manifold with positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained from the toric diagrams with any height, or equivalently on all compact toric Sasaki manifolds whose cones have flat canonical bundle. We further show that there exists an infinite family of inequivalent toric Sasaki-Einstein metrics on \(S^5 \sharp k(S^2 \times S^3)\) for each positive integer k.

On some metrics compatible with the Fell–Matheron topology

Modifying the Hausdorff–Buseman metric, we obtain a compatible metric with the Fell–Matheron topology on the space of closed subsets of a locally compact Hausdorff second countable space. We also give an alternative expression of this metric, and two more compatible metrics, a metric of summation form and a modified Rockafellar–Wets metric. Through the study of the relation between the original Hausdorff–Buseman metric and the Rockafellar–Wets metric, it is also shown that the convergence in the original Hausdorff–Buseman metric in the space of non-empty closed subsets of R N is equivalent to the Painlevé–Kuratowski convergence.

A Metamathematical Condition Equivalent to the Existence of a Complete Left Invariant Metric for a Polish Group

AbstractStrengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants. Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.

How many miles to βX?— [formula omitted] miles, or just one foot

It is known that the Stone–Čech compactification βX of a metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X. If we confine ourselves to locally compact separable metrizable spaces, the corresponding statement holds for Higson compactifications. We investigate the smallest cardinality of a set D of compatible metrics on X such that βX is approximated by Smirnov or Higson compactifications for all metrics in D. We prove that it is either the dominating number or 1 for a locally compact separable metrizable space.

Conformally flat pencils of metrics, Frobenius structures and a modified Saito construction

The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors’ earlier work [L. David, I.A.B. Strachan, Compatible metrics on manifolds and non-local bi-Hamiltoninan structures, Int. Math. Res. Notices 66 (2004) 3533–3557], much of the structure comes from the compatibility property of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of metrics are studied and examples, based on a modification of the Saito construction, are developed.

Topological entropy for divergence points

Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure as\[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \]where is continuous and affine with respect to the weak topology on M(X). We think of the composite\[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \]as a continuous and affine deformation of the empirical measure Ln. The set of divergence points of such a deformation is defined as\[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \]In this paper we show that for a continuous transformation satisfying the specification property, if is not a singleton, then the set of divergence points has full topological entropy, i.e.\[ h_{\rm top}(D(f,\Xi))=h_{\rm top}(f). \]

How many miles to βω?—Approximating βω by metric-dependent compactifications

It is known that the Stone–Čech compactification βX of a noncompact metrizable space X is approximated by the collection of Smirnov compactifications of X for all compatible metrics on X. We investigate the smallest cardinality of a set D of compatible metrics on the countable discrete space ω such that, βω is approximated by Smirnov compactifications for all metrics in D, but any finite subset of D does not suffice. We also study the corresponding cardinality for Higson compactifications.

Probability theory in fuzzy sample spaces

This paper tries to develop a neat and comprehensive probability theory for sample spaces where the events are fuzzy subsets of Open image in new window The investigations are focussed on the discussion how to equip those sample spaces with suitable σ-algebras and metrics. In the end we can point out a unified concept of random elements in the sample spaces under consideration which is linked with compatible metrics to express random errors. The result is supported by presenting a strong law of large numbers, a central limit theorem and a Glivenko-Cantelli theorem for these kinds of random elements, formulated simultaneously w.r.t. the selected metrics. As a by-product the line of reasoning, which is followed within the paper, enables us to generalize as well as to bring together already known results and concepts from literature.

Compatible metrics on a manifold and nonlocal bi-Hamiltonian structures

Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated non-local bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasi-homogeneity assumptions one arrives at a so-called weak $\F$-manifold - a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.

Poisson manifolds with compatible pseudo-metric and pseudo-Riemannian Lie algebras

In a previous paper (C. R. Acad. Sci. Paris Sér. I 333 (2001) 763–768), the author introduced a notion of compatibility between a Poisson structure and a pseudo-Riemannian metric. In this paper, we introduce a new class of Lie algebras called pseudo-Riemannian Lie algebras. The two notions are closely related: we prove that the dual of a Lie algebra endowed with its canonical linear Poisson structure carries a compatible pseudo-Riemannian metric if and only if the Lie algebra is a pseudo-Riemannian Lie algebra. Moreover, the Lie algebra obtained by linearizing at a point a Poisson manifold with a compatible pseudo-Riemannian metric is a pseudo-Riemannian Lie algebra. We also give some properties of the symplectic leaves of such manifolds, and we prove that every Poisson manifold with a compatible Riemannian metric is unimodular. Finally, we study Poisson Lie groups endowed with a compatible pseudo-Riemannian metric, and we give the classification of all pseudo-Riemannian Lie algebras of dimension 2 and 3.

Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.

Is the composite of two expansive maps expansive?

We study the question whether the composite of two expansive maps is itself expansive. The answer to this question is a resounding no. We will actually establish a stronger fact. We will give an example of two dynamical systems which are topologically exact and Ruelle expanding with respect to two different, equivalent and compatible metrics, but whose composite is far from being expansive, for it is locally eventually contracting towards a fixed point a completely invariant, dense open subspace.

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 Dk(X,Y) the topology of uniform convergence on compact sets on D(X,Y). We prove the following main results: Dk(X,Y) is metrizable iff Dk(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 Dk(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.

Recognizing special metrics by topological properties of the ``metri''-proximal hyperspace

In this paper, we first characterize those compatible metrics $d$ on a metrizable space $X$ which give rise to a connected $d$-proximal hyperspace. We show that the space of irrational numbers, in particular, admits a complete metric with this property and, as a consequence, we get a negative answer to a question of [11] about selections for hyperspace topologies. Next, we characterize the compatible metrics on $X$ which are uniformly equivalent to ultrametrics showing that this is equivalent to the zero-dimensionality of the corresponding proximal hyperspaces. Applications and related results about other disconnectedness-like properties of proximal hyperspaces are obtained.

Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability

The description problem is solved for compatible metrics of constant Riemannian curvature. Nonlinear equations describing all nonsingular pencils of compatible metrics of constant Riemannian curvature are derived and their integrability by the inverse scattering method is proved. In particular, a Lax pair with a spectral parameter is found for these nonlinear equations. We prove that all nonsingular pairs of compatible metrics of constant Riemannian curvature are described by special integrable reductions of the nonlinear equations defining orthogonal curvilinear coordinate systems in spaces of constant curvature.

INTEGRABILITY OF THE EQUATIONS FOR NONSINGULAR PAIRS OF COMPATIBLE FLAT METRICS

We solve the problem of describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) in the general N-component case. This problem is equivalent to the problem of describing all compatible Dubrovin–Novikov brackets (compatible nondegenerate local Poisson brackets of hydrodynamic type) playing an important role in the theory of integrable systems of hydrodynamic type and also in modern differential geometry and field theory. We prove that all nonsingular pairs of compatible flat metrics are described by a system of nonlinear differential equations that is a special nonlinear differential reduction of the classical Lame equations, and we present a scheme for integrating this system by the method of the inverse scattering problem. The integration procedure is based on using the Zakharov method for integrating the Lame equations (a version of the inverse scattering method).

Compatible flat metrics

We solve the problem of description of nonsingular pairs of compatible flat metrics for the general N‐component case. The integrable nonlinear partial differential equations describing all nonsingular pairs of compatible flat metrics (or, in other words, nonsingular flat pencils of metrics) are found and integrated. The integrating of these equations is based on reducing to a special nonlinear differential reduction of the Lamé equations and using the Zakharov method of differential reductions in the dressing method (a version of the inverse scattering method).