A new characterization for isometries by triangles
نویسندگان
چکیده
Let R be an n-dimensional Euclidean space and D be an n-dimensional hyperbolic space with the Poincaré metric for n > 1. In this paper, we shall prove the following results. (i) A bijection f : D → D n is an isometry (Möbius transformation) if and only if f is triangle preserving. (ii) A bijection f : R → R is an affine transformation if and only if f is triangle preserving.
منابع مشابه
On Tensor Product of Graphs, Girth and Triangles
The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.
متن کاملOn Commutators of Isometries and Hyponormal Operators
A sufficient condition is obtained for two isometries to be unitarily equivalent. Also, a new class of M-hyponormal operator is constructed
متن کاملDegenerations of Representations and Thin Triangles
This paper gives a compactification of the space of representations of a finitely generated group into the groups of isometries of all spaces with ∆-thin triangles. The ideal points are actions on R-trees. It is a geometric reformulation and extension of the Culler-Morgan-Shalen theory concerning limits of (characters of) representations into SL2C and more generally SO(n, 1).
متن کاملComplete Isometries - an Illustration of Noncommutative Functional Analysis
This article, addressed to a general audience of functional analysts, is intended to be an illustration of a few basic principles from ‘noncommutative functional analysis’, more specifically the new field of operator spaces. In our illustration we show how the classical characterization of (possibly non-surjective) isometries between function algebras generalizes to operator algebras. We give s...
متن کاملAlgebraic Characterization of the Isometries of the Complex and Quaternionic Hyperbolic Plane
in terms of their trace and determinant are foundational in the real hyperbolic geometry. The counterpart of this characterization for isometries of H C was given by Giraud [8] and Goldman [9]. In this paper we offer algebraic characterization for the isometries of H H. The methods we follow carry over to the complex hyperbolic space, and yields an alternative characterization of the isometries...
متن کامل