In mathematics, the isometry group of a metric space is the set of all bijective isometries (that is, bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function. The elements of the isometry group are sometimes called motions of the space.
Every isometry group of a metric space is a subgroup of isometries. It represents in most cases a possible set of symmetries of objects/figures in the space, or functions defined on the space. See symmetry group.
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set.
In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form; transformations preserving this form are sometimes called "isometries", and the collection of them is then said to form an isometry group of the pseudo-Euclidean space.
Examples
- The isometry group of the subspace of a metric space consisting of the points of a scalene triangle is the trivial group. A similar space for an isosceles triangle is the cyclic group of order two, C2. A similar space for an equilateral triangle is D3, the dihedral group of order 6.
- The isometry group of a two-dimensional sphere is the orthogonal group O(3).
- The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).
- The isometry group of the Poincaré disc model of the hyperbolic plane is the projective special unitary group PSU(1,1).
- The isometry group of the Poincaré half-plane model of the hyperbolic plane is PSL(2,R).
- The isometry group of Minkowski space is the Poincaré group.
- Riemannian symmetric spaces are important cases where the isometry group is a Lie group.
See also
- Point group
- Point groups in two dimensions
- Point groups in three dimensions
- Fixed points of isometry groups in Euclidean space
References
- Roman, Steven (2008), Advanced Linear Algebra, Graduate Texts in Mathematics (Third ed.), Springer, p. 271, ISBN 978-0-387-72828-5.
- Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001), A course in metric geometry, Graduate Studies in Mathematics, vol. 33, Providence, RI: American Mathematical Society, p. 75, ISBN 0-8218-2129-6, MR 1835418.
- Berger, Marcel (1987), Geometry. II, Universitext, Berlin: Springer-Verlag, p. 281, doi:10.1007/978-3-540-93816-3, ISBN 3-540-17015-4, MR 0882916.
- Olver, Peter J. (1999), Classical invariant theory, London Mathematical Society Student Texts, vol. 44, Cambridge: Cambridge University Press, p. 53, doi:10.1017/CBO9780511623660, ISBN 0-521-55821-2, MR 1694364.
- Müller-Kirsten, Harald J. W.; Wiedemann, Armin (2010), Introduction to supersymmetry, World Scientific Lecture Notes in Physics, vol. 80 (2nd ed.), Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., p. 22, doi:10.1142/7594, ISBN 978-981-4293-42-6, MR 2681020.
In mathematics the isometry group of a metric space is the set of all bijective isometries that is bijective distance preserving maps from the metric space onto itself with the function composition as group operation Its identity element is the identity function The elements of the isometry group are sometimes called motions of the space Every isometry group of a metric space is a subgroup of isometries It represents in most cases a possible set of symmetries of objects figures in the space or functions defined on the space See symmetry group A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set In pseudo Euclidean space the metric is replaced with an isotropic quadratic form transformations preserving this form are sometimes called isometries and the collection of them is then said to form an isometry group of the pseudo Euclidean space ExamplesThe isometry group of the subspace of a metric space consisting of the points of a scalene triangle is the trivial group A similar space for an isosceles triangle is the cyclic group of order two C2 A similar space for an equilateral triangle is D3 the dihedral group of order 6 The isometry group of a two dimensional sphere is the orthogonal group O 3 The isometry group of the n dimensional Euclidean space is the Euclidean group E n The isometry group of the Poincare disc model of the hyperbolic plane is the projective special unitary group PSU 1 1 The isometry group of the Poincare half plane model of the hyperbolic plane is PSL 2 R The isometry group of Minkowski space is the Poincare group Riemannian symmetric spaces are important cases where the isometry group is a Lie group See alsoPoint group Point groups in two dimensions Point groups in three dimensions Fixed points of isometry groups in Euclidean spaceReferencesRoman Steven 2008 Advanced Linear Algebra Graduate Texts in Mathematics Third ed Springer p 271 ISBN 978 0 387 72828 5 Burago Dmitri Burago Yuri Ivanov Sergei 2001 A course in metric geometry Graduate Studies in Mathematics vol 33 Providence RI American Mathematical Society p 75 ISBN 0 8218 2129 6 MR 1835418 Berger Marcel 1987 Geometry II Universitext Berlin Springer Verlag p 281 doi 10 1007 978 3 540 93816 3 ISBN 3 540 17015 4 MR 0882916 Olver Peter J 1999 Classical invariant theory London Mathematical Society Student Texts vol 44 Cambridge Cambridge University Press p 53 doi 10 1017 CBO9780511623660 ISBN 0 521 55821 2 MR 1694364 Muller Kirsten Harald J W Wiedemann Armin 2010 Introduction to supersymmetry World Scientific Lecture Notes in Physics vol 80 2nd ed Hackensack NJ World Scientific Publishing Co Pte Ltd p 22 doi 10 1142 7594 ISBN 978 981 4293 42 6 MR 2681020
