ISOMETRIES OF Rn
نویسنده
چکیده
An isometry of Rn is a function h : Rn → Rn that preserves the distance between vectors: ||h(v)− h(w)|| = ||v − w|| for all v and w in Rn, where ||(x1, . . . , xn)|| = √ x1 + · · ·+ xn. Example 1.1. The identity transformation: id(v) = v for all v ∈ Rn. Example 1.2. Negation: − id(v) = −v for all v ∈ Rn. Example 1.3. Translation: fixing u ∈ Rn, let tu(v) = v + u. Easily ||tu(v) − tu(w)|| = ||v − w||.
منابع مشابه
A Remark on Quasi-isometries
We show that if f : Bn -IRn is an e-quasi-isometry, with e < 1, defined on the unit ball Bn of Rn, then there is an affine isometry h : Bn -Rn with lIf(x) -h(x)|I < Ce(l+logn) where C is a universal constant. This result is sharp.
متن کاملQuasi-isometries and Rigidity of Solvable Groups
In this note, we announce the first results on quasi-isometric rigidity of non-nilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasi-isometric to R⋉Rn where the semidirect product is defined by a diagonalizable matrix of determinant one with ...
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 کاملSome properties of Diametric norm and its isometries on B_0 (Bbb R)$
This article has no abstract.
متن کاملTheta functions and weighted theta functions of Euclidean lattices, with some applications
By “Euclidean space” of dimension n we mean a real vector space of dimension n, equipped with a positive-definite inner product 〈·, ·〉. We usually call such a space “Rn” even when there is no distinguished choice of coordinates. A lattice in Rn is a discrete co-compact subgroup L ⊂ Rn, that is, a discrete subgroup such that the quotient Rn/L is compact (and thus necessarily homeomorphic with th...
متن کامل