Related papers: Metric transforms yielding Gromov hyperbolic space…

In the first part of this investigation, [Ha], we generalized a weighted distance function of [Li] and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are…

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian spaces $(M=G/H,g)$ whose geodesics are orbits of one-parameter subgroups of $G$. The corresponding metric $g$ is called a geodesic orbit metric. We study the…

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it geodesic triangle} $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-\emph{hyperbolic} $($in the…

Let $\Phi$ be a strictly plurisubharmonic and radial function on the unit disk ${\cal D}\subset {\complex}$ and let $g$ be the \K metric associated to the \K form $\omega =\frac{i}{2}\partial\bar\partial\Phi$. We prove that if $g$ is…

Given a compact orientable surface with finitely many punctures $\Sigma$, let $\Cal S(\Sigma)$ be the set of isotopy classes of essential unoriented simple closed curves in $\Sigma$. We determine a complete set of relations for a function…

Many concrete problems are formulated in terms of a finite set of points in $R^n$ which, via the ambient Euclidean metric, becomes a finite metric space. To obtain information from such a space, it is often useful to associate a graph to…

In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov…

This thesis is concerned with extending the idea of geodesic completeness from pseudo-Riemannian to complex geometry: we take, however a completely holomorphicpoint of view; that is to say, a 'metric' will be a (meromorphic) symmetric…

Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This…

We investigate the limiting behaviour of the path of random bridges treated as random sets in $\mathbb{R}^{d}$ with the Euclidean metric and the dimension $d$ increasing to infinity. The main result states that, in the square integrable…

A Finsler metric is geodesically reversible if geodesics remain geodesics after a change of orientation. Asymmetric norms on vector spaces and Funk metrics in the interior of convex bodies are examples of geodesically reversible metrics…

We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space $(X, d)$ is ultrametric iff the diametrical graph of the metric $d_{\varepsilon}(x,…

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to…

We show that for every nondecreasing concave function w:R+ --> R+ with w(0)=0, either every finite metric space embeds with distortion arbitrarily close to 1 into a metric space of the form (X,w o d) for some metric d on X, or there exists…

The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.

On a smooth connected manifold, we consider all possible locally elliptic and locally bounded measurable coefficient Riemannian metrics called rough Riemannian metrics. We equip this set with an extended metric which is connected if and…

Conformal transformations of a Euclidean (complex) plane have some kind of completeness (sufficiency) for the solution of many mathematical and physical-mathematical problems formulated on this plane. There is no such completeness in the…

We present an adaptive geometry in which the yardstick co-deforms with space itself, formulated on cellular spaces where length is a count: distances are shortest cell-crossing counts. No cell shape, angles, or embedding are assumed; the…

This paper concerns the elastic structures which exhibit non-zero strain at free equilibria. Many growing tissues (leaves, flowers or marine invertebrates) attain complicated configurations during their free growth. Our study departs from…

In many real-world applications data come as discrete metric spaces sampled around 1-dimensional filamentary structures that can be seen as metric graphs. In this paper we address the metric reconstruction problem of such filamentary…