Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the identity function.[1] 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).[2]

• The isometry group of the n-dimensional Euclidean space is the Euclidean group E(n).[3]
• The isometry group of the Poincaré disc model of the hyperbolic plane is the projective special unitary group SU(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.[4]

• Riemannian symmetric spaces are important cases where the isometry group is a Lie group.

See also

• Point groups in two dimensions
• Point groups in three dimensions
• Fixed points of isometry groups in Euclidean space

References

1. Burago, Dmitri; Burago, Yuri; Ivanov, Sergei (2001), 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. 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.
3. Olver, Peter J. (1999), 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.
4. Müller-Kirsten, Harald J. W.; Wiedemann, Armin (2010), Introduction to supersymmetry, World Scientific Lecture Notes in Physics, 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.