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

Introduction In many diffusion tensor imaging (DTI) analysis methods, including registration, realignment and re-slicing, averaging or interpolating tensors is required. Defining how distance is measured between tensors, through a metric, determines the interpolation results. It was demonstrated that when using a conventional Euclidean metric, the resulted tensor might have a larger volume (det...

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...

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 the topology of metric spaces which are definable in o-minimal expansions of ordered fields. We show that a definable metric space either contains an infinite definable discrete set or is definably homeomorphic to a definable set equipped with its euclidean topology. This implies that a separable metric space which is definable in an o-minimal expansion of the real field is definably h...

The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (σ-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of triangle ABR vanishes. The triangle ar...