Grup de izometrie

În matematică grupul de izometrie al unui spațiu metric este mulțimea tuturor izometriilor bijective (adică aplicații bijective, care păstrează distanța) din spațiul metric pe el însuși, cu compunerea funcțiilor⁠(d) ca operație de grup. Elementul său neutru este funcția identitate.[1] Elementele grupului de izometrie sunt uneori numite deplasări[2] ale spațiului.

Fiecare grup de izometrie al unui spațiu metric este un subgrup de izometrii. Reprezintă în cele mai multe cazuri un posibil set de simetrii ale obiectelor/figurilor din spațiu, sau ale funcțiilor definite pe spațiu. Vezi grup de simetrie.

Un grup de izometrie discret este un grup de izometrii astfel încât pentru fiecare punct al spațiului setul de imagini ale punctului din izometrii este o mulțime discretă.

În spațiul pseudoeuclidian⁠(d) metrica este înlocuită cu o formă cvadratică izotropă⁠(d); transformările care conservă această formă sunt uneori numite „izometrii”, iar mulțimea lor se spune că formează un grup de izometrie al spațiului pseudoeuclidian.

Exemple

Note

  1. ^ en Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (), A course in metric geometry, Graduate Studies in Mathematics, 33, Providence, RI: American Mathematical Society, p. 75, ISBN 0-8218-2129-6, MR 1835418 
  2. ^ V. Popescu, Geometrie analitică[nefuncțională], pub.ro, accesat 2022-02-05, p. 146
  3. ^ en Berger, Marcel (), Geometry. II, Universitext, Berlin: Springer-Verlag, p. 281, doi:10.1007/978-3-540-93816-3, ISBN 3-540-17015-4, MR 0882916 .
  4. ^ en Olver, Peter J. (), Classical invariant theory, London Mathematical Society Student Texts, 44, Cambridge: Cambridge University Press, p. 53, doi:10.1017/CBO9780511623660, ISBN 0-521-55821-2, MR 1694364 .
  5. ^ en Müller-Kirsten, Harald J. W.; Wiedemann, Armin (), Introduction to supersymmetry, World Scientific Lecture Notes in Physics, 80 (ed. 2nd), Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., p. 22, doi:10.1142/7594, ISBN 978-981-4293-42-6, MR 2681020 .
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