Isometry group
{{Short description|Automorphism group of a metric space or pseudo-Euclidean space}}
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.{{citation |last=Roman |first=Steven |title=Advanced Linear Algebra |date=2008 |pages=271 |series=Graduate Texts in Mathematics |edition=Third |publisher=Springer |isbn=978-0-387-72828-5 |author-link=Steven Roman}}. Its identity element is the identity function.{{citation
| last1 = Burago | first1 = Dmitri
| last2 = Burago | first2 = Yuri
| last3 = Ivanov | first3 = Sergei
| isbn = 0-8218-2129-6
| mr = 1835418
| page = 75
| publisher = American Mathematical Society
| location = Providence, RI
| series = Graduate Studies in Mathematics
| title = A course in metric geometry
| url = https://books.google.com/books?id=afnlx8sHmQIC&pg=PA75
| volume = 33
| year = 2001}}. 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).{{citation
| last = Berger | first = Marcel
| doi = 10.1007/978-3-540-93816-3
| isbn = 3-540-17015-4
| mr = 882916
| page = 281
| publisher = Springer-Verlag
| location = Berlin
| series = Universitext
| title = Geometry. II
| url = https://books.google.com/books?id=6WZHAAAAQBAJ&pg=PA281
| year = 1987}}.
- The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).{{citation
| last = Olver | first = Peter J. |author-link=Peter J. Olver
| doi = 10.1017/CBO9780511623660
| isbn = 0-521-55821-2
| mr = 1694364
| page = 53
| publisher = Cambridge University Press
| location = Cambridge
| series = London Mathematical Society Student Texts
| title = Classical invariant theory
| url = https://books.google.com/books?id=1GlHYhNRAqEC&pg=PA53
| volume = 44
| year = 1999}}.
- 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.{{citation
| last1 = Müller-Kirsten | first1 = Harald J. W.
| last2 = Wiedemann | first2 = Armin
| doi = 10.1142/7594
| edition = 2nd
| isbn = 978-981-4293-42-6
| mr = 2681020
| page = 22
| publisher = World Scientific Publishing Co. Pte. Ltd.
| location = Hackensack, NJ
| series = World Scientific Lecture Notes in Physics
| title = Introduction to supersymmetry
| url = https://books.google.com/books?id=RU-hsrWp9isC&pg=PA22
| volume = 80
| year = 2010}}.
- Riemannian symmetric spaces are important cases where the isometry group is a Lie group.
See also
References
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