We axiomatize the class of groups generated by the pointreflections of a metric plane with a non-Euclidean metric, the structure of which turns out to be very rich compared to the Euclidean metric case, and state an open problem. MSC 2000: 51M05, 51M10, 51F15, 03B30

It is proved the following theorem, if w is a quasiconformal harmonic mappings between two Riemann surfaces with smooth boundary and aproximate analytic metric, then w is a quasi-isometry with respect to Euclidean metric.

We investigate connections between pairs of Riemannian metrics whose sum is a (tensor) product of a covector field with itself. As a special result is constructed one-to-one mapping between the classes of Euclidean and Lorentzian metrics. The existence of Lorentzian metrics on a differentiable manifold is discussed. We point the possibility that any physical theory based on Lorentzian metric(s)...

Riemannian manifolds have been widely employed for video representations in visual classification tasks including videobased face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the non-linear geometry of the underlying Riemannian manifolds. In this paper, we propose a novel metric learning framework to learn a distance metric across a Euclid...

The Minimum Weight Euclidean Matching (MWEM) problem is: given 2n point sites in the plane with Euclidean metric for interpoint distances, match the sites into n pairs so that the sum of the n distances between matched pairs is minimized. The graph theoretic version of this problem has been extensively studied since the pioneering work of Edmonds. The best time bound known for MEWM is O(n 2:5 (...

A greedy embedding of an unweighted undirected graph G = (V, E) into a metric space (X, ρ) is a function f : V → X such that for every source-sink pair of different vertices s, t ∈ V it is the case that s has a neighbor v in G with ρ(f(v), f(t)) < ρ(f(s), f(t)). Finding greedy embeddings of connectivity graphs helps to build distributed routing schemes with compact routing tables. In this paper...

As introduced by Weaver, (metric) derivations extend the notion of differential operators on Euclidean spaces and provide a linear differentiation theory that is well-defined on all metric spaces with Borel measures. The main result of this paper is a rigidity theorem for measures on Euclidean spaces that support a maximal number of derivations. The proof relies substantially on ideas from geom...

We prove that every finite-volume hyperbolic 3-manifold M with p ≥ 1 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K∗ M . Moreover: (a) The universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry; (...

We study Euclidean embeddings of Euclidean metrics and present the following four results: (1) an O(log n √ log logn) approximation for minimum bandwidth in conjunction with a semi-definite relaxation, (2) an O(log n) approximation in O(n) time using a new constraint set, (3) a lower bound of Θ( √ logn) on the least possible volume distortion for Euclidean metrics, (4) a new embedding with O( √...