共查询到20条相似文献,搜索用时 31 毫秒
1.
Valerio Capraro 《Expositiones Mathematicae》2013,31(4):334-349
We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that many spaces with uniform bounded geometry having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces. 相似文献
2.
We establish a general slice theorem for the action of a locally convex Lie group on a locally convex manifold, which generalizes the classical slice theorem of Palais to infinite dimensions. We discuss two important settings under which the assumptions of this theorem are fulfilled. First, using Glöckner's inverse function theorem, we show that the linear action of a compact Lie group on a Fréchet space admits a slice. Second, using the Nash–Moser theorem, we establish a slice theorem for the tame action of a tame Fréchet Lie group on a tame Fréchet manifold. For this purpose, we develop the concept of a graded Riemannian metric, which allows the construction of a path-length metric compatible with the manifold topology and of a local addition. Finally, generalizing a classical result in finite dimensions, we prove that the existence of a slice implies that the decomposition of the manifold into orbit types of the group action is a stratification. 相似文献
3.
There are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular
basepoint (a complex or a hyperbolic structure on the surface). Such spaces include the quasiconformal Teichmüller space,
the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces
are set-theoretically different. An important question is therefore to understand relations between them. Each of these spaces
is equipped with its own metric, and under some hypotheses, there are inclusions between them. In this paper, we obtain local
metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain
hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of
finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the
case of a surface of finite type with no boundary components (but possibly with punctures), we show that the restriction of
the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric
on the domain and the classical Teichmüller metric on the range. In the case of a surface of finite type with punctures and
boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous
to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that
the restriction of the identity map to any “relative thick” part of Teichmüller space is globally bi-Lipschitz, with respect
to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the
range. 相似文献
4.
We show that an adaptation of the augmenting path method for graphs proves Menger’s Theorem for wide classes of topological
spaces. For example, it holds for locally compact, locally connected, metric spaces, as already known. The method lends itself
particularly well to another class of spaces, namely the locally arcwise connected, hereditarily locally connected, metric
spaces. Finally, it applies to every space where every point can be separated from every closed set not containing it by a
finite set, in particular to every subspace of the Freudenthal compactification of a locally finite, connected graph. While
closed subsets of such a space behave nicely in that they are compact and locally connected (and therefore locally arcwise
connected), the general subspaces do not: They may be connected without being arcwise connected. Nevertheless, they satisfy
Menger’s Theorem.
This work was carried out while Antoine Vella was a Marie Curie Fellow at the Technical University of Denmark, as part of
the research project TOPGRAPHS (Contract MEIF-CT-2005-009922), under the supervision of Carsten Thomassen. 相似文献
5.
Ayşe Sönmez 《Applied Mathematics Letters》2010,23(4):494-497
It is well known that any metric space is paracompact. As a generalization of metric spaces, cone metric spaces play very important role in fixed point theory, computer science, and some other research areas as well as in general topology. In this paper, a theorem which states that any cone metric space with a normal cone is paracompact is proved. 相似文献
6.
J.W. Cannon 《Topology and its Applications》2006,153(14):2648-2672
We study here a number of questions raised by examining the fundamental groups of complicated one-dimensional spaces. The first half of the paper considers one-dimensional spaces as such. The second half proves related results for general spaces that are needed in the first half but have independent interest. Among the results we prove are the theorem that the fundamental group of a separable, connected, locally path connected, one-dimensional metric space is free if and only if it is countable if and only if the space has a universal cover and the theorem that the fundamental group of a compact, one-dimensional, connected metric space embeds in an inverse limit of finitely generated free groups and is shape injective. 相似文献
7.
《Fuzzy Sets and Systems》2004,146(1):121-133
In this paper, we show that weakly null-additive fuzzy measures on metric spaces possess regularity. Lusin's theorem, which is well-known in classical measure theory, is generalized to fuzzy measure space by using the regularity and weakly null-additivity. A version of Egoroff's theorem for the fuzzy measure defined on metric spaces is given. An application of Lusin's theorem to approximation in the mean of measurable function on fuzzy measure spaces is presented. 相似文献
8.
Stefan Wenger 《Calculus of Variations and Partial Differential Equations》2007,28(2):139-160
In compact local Lipschitz neighborhood retracts in
weak convergence for integral currents is equivalent to convergence with respect to the flat distance. This comes as a consequence of the deformation theorem for currents in Euclidean space. Working in the setting of metric integral currents (the theory of which was developed by Ambrosio and Kirchheim) we prove that the equivalence of weak and flat convergence remains true in the more general context of metric spaces admitting local cone type inequalities. These include in particular all Banach spaces and all CAT(κ)-spaces. As an application we obtain the existence of a minimal element in a fixed homology class and show that the weak limit of a sequence of minimizers is itself a minimizer. 相似文献
9.
John Kulesza Teck-Cheong Lim 《Proceedings of the American Mathematical Society》1996,124(11):3345-3349
We prove that weak compactness and countable weak compactness in metric spaces are not equivalent. However, if the metric space has normal structure, they are equivalent. It follows that some fixed point theorems proved recently are consequences of a classical theorem of Kirk.
10.
Stefan Wenger 《Calculus of Variations and Partial Differential Equations》2011,40(3-4):423-448
By Gromov??s compactness theorem for metric spaces, every uniformly compact sequence of metric spaces admits an isometric embedding into a common compact metric space in which a subsequence converges with respect to the Hausdorff distance. Working in the class of oriented k-dimensional Riemannian manifolds (with boundary) and, more generally, integral currents in metric spaces in the sense of Ambrosio?CKirchheim and replacing the Hausdorff distance with the filling volume or flat distance, we prove an analogous compactness theorem in which however we only assume uniform bounds on volume and diameter. 相似文献
11.
Valera Berestovskii 《Topology and its Applications》2007,154(8):1748-1777
We develop a generalized covering space theory for a class of uniform spaces called coverable spaces. Coverable spaces include all geodesic metric spaces, connected and locally pathwise connected compact topological spaces, in particular Peano continua, as well as more pathological spaces like the topologist's sine curve. The uniform universal cover of a coverable space is a kind of generalized cover with universal and lifting properties in the category of uniform spaces and uniformly continuous mappings. Associated with the uniform universal cover is a functorial uniform space invariant called the deck group, which is related to the classical fundamental group by a natural homomorphism. We obtain some specific results for one-dimensional spaces. 相似文献
12.
We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn’s Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces. 相似文献
13.
We prove a version of the Hopf–Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf–Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure. 相似文献
14.
Li Zezhi 《数学年刊B辑(英文版)》1984,5(2):135-140
Two existence theorems of random measures on a separable complete metric space areproved.It seems that the theory of random measures in locally compact spaces and that inseparable complete metric spaces are essentially different by noting that the critera fortightness of the locally finite measures is much more tedious than that of the Radommeasures. 相似文献
15.
We give an elementary proof of the classical Menger result according to which any metric space X that consists of more than four points is isometrically imbedded into
\mathbbR \mathbb{R} if every three-point subspace of X is isometrically imbedded into
\mathbbR \mathbb{R} . A series of corollaries of this theorem is obtained. We establish new criteria for finite metric spaces to be isometrically
imbedded into
\mathbbR \mathbb{R} . 相似文献
16.
A note on closed images of locally compact metric spaces 总被引:1,自引:0,他引:1
Shou Lin 《Acta Mathematica Hungarica》2005,109(1-2):157-162
Summary A decomposition theorem about closed images of locally compact metric spaces is discussed. It is shown that a space is a closed
image of a locally compact metric space if and only if it is a regular Fréchet space with a point-countable k-network, and each of its closed first-countable subset is locally compact. 相似文献
17.
Sumio Yamada 《Geometriae Dedicata》2010,145(1):43-63
On a Teichmüller space, the Weil-Petersson metric is known to be incomplete. Taking metric and geodesic completions result
in two distinct spaces, where the Hopf-Rinow theorem is no longer relevant due to the singular behavior of the Weil-Petersson
metric. We construct a geodesic completion of the Teichmüller space through the formalism of Coxeter complex with the Teichmüller
space as its non-linear non-homogeneous fundamental domain. We then show that the metric and geodesic completions both satisfy
a finite rank property, demonstrating a similarity with the non-compact symmetric spaces of semi-simple Lie groups. 相似文献
18.
Michael Temkin 《Israel Journal of Mathematics》2011,185(1):1-42
In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical
Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either
as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these
spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition
theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition
of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem. 相似文献
19.
Cs. Vincze 《Differential Geometry and its Applications》2006,24(1):1-20
As it is well-known, a Minkowski space is a finite dimensional real vector space equipped with a Minkowski functional F. By the help of its second order partial derivatives we can introduce a Riemannian metric on the vector space and the indicatrix hypersurface S:=F−1(1) can be investigated as a Riemannian submanifold in the usual sense.Our aim is to study affine vector fields on the vector space which are, at the same time, affine with respect to the Funk metric associated with the indicatrix hypersurface. We give an upper bound for the dimension of their (real) Lie algebra and it is proved that equality holds if and only if the Minkowski space is Euclidean. Criteria of the existence is also given in lower dimensional cases. Note that in case of a Euclidean vector space the Funk metric reduces to the standard Cayley-Klein metric perturbed with a nonzero 1-form.As an application of our results we present the general solution of Matsumoto's problem on conformal equivalent Berwald and locally Minkowski manifolds. The reasoning is based on the theory of harmonic vector fields on the tangent spaces as Riemannian manifolds or, in an equivalent way, as Minkowski spaces. Our main result states that the conformal equivalence between two Berwald manifolds must be trivial unless the manifolds are Riemannian. 相似文献
20.
We study a metric version of the simplicial volume on Riemannianmanifolds, the Lipschitz simplicial volume, with applicationsto degree theorems in mind. We establish a proportionality principleand a product inequality from which we derive an extension ofGromov's volume comparison theorem to products of negativelycurved manifolds or locally symmetric spaces of noncompact type.In contrast, we provide vanishing results for the ordinary simplicialvolume; for instance, we show that the ordinary simplicial volumeof noncompact locally symmetric spaces with finite volume of-rank at least 3 is zero. Received November 6, 2007. Revised August 20, 2008. 相似文献