Related papers: An intrinsic characterization of five points in a …

We show that any metric on $S^2$ with Gauss curvature $K \geq -\kappa$ admits a $C^{1,1}$-isometric embedding into the hyperbolic space with sectional curvature $-\kappa$. We also give a sufficient condition for a metric on $S^2$ to be…

Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real…

The five-dimensional (5D) Riemannian G\"odel-type manifolds are examined in light of the equivalence problem techniques, as formulated by Cartan. The necessary and sufficient conditions for local homogeneity of these 5D manifolds are…

We show that for n > 4 a length space (X,d) satisfies a rough n-point condition if and only it it is rough CAT(0).

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space…

Richmond and Richmond (Amer. Math. Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove…

If a class of finitely generated groups Curly(G) is closed under isometric amalgamations along free subgroups, then every G in Curly(G) can be quasi-isometrically embedded in a group Hat(G) in Curly(G) that has no proper subgroups of finite…

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two…

Necessary and sufficient quantitative geometric conditions are given for an unbounded set A in a euclidean space R^n to have the following property with a given c > 0: For every s > 0 and for every s-nearisometry f: A -> R^n there is an…

Under certain conditions, a $(1+1)$-dimensional slice $\hat{g}$ of a spherically symmetric black hole spacetime can be equivariantly embedded in $(2+1)$-dimensional Minkowski space. The embedding depends on a real parameter that corresponds…

A spacetime is a connected 4-dimensional semi-Riemannian manifold endowed with a metric $g$ with signature $(- + + +)$. The geometry of a spacetime is described by the metric tensor $g$ and the Ricci tensor $S$ of type $(0, 2)$ whereas the…

We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We…

For any intrinsic Gromov hyperbolic space we establish a Gehring-Hayman type theorem for conformally deformed spaces. As an application, we prove that any complete intrinsic hyperbolic space with atleast two points in the Gromov boundary…

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C…

In 1996 A. Alexandrov solved an isometric embedding problem for model spaces $K_\Theta$ with an arbitrary inner function $\Theta$. We find all extreme points of this convex set of measures in the case when $\Theta$ is a finite Blaschke…

The Grushin spaces, as one of the most important models in the Carnot-Carath\'eodory space, are a class of locally compact and geodesic metric spaces which admit a dilation. Function spaces on Grushin spaces and some related geometric…

We prove an inequality concerning isometries of a Gromov hyperbolic metric space, which does not require the space to be proper or geodesic. It involves the joint stable length, a hyperbolic version of the joint spectral radius, and shows…

A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as…

We study in the setting of a metric space $\left( X,d\right) $ some generalizations of four hyperbolic-type metrics defined on open sets $G$ with nonempty boundary in the $n-$dimensional Euclidean space, namely Gehring-Osgood metric,…

In this paper, using the existence of infinite equidistant subsets of closed balls, we characterize the injectivity of ultrametric spaces for finite ultrametric spaces, which also gives a characterization of the Urysohn universal…