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.

- 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, C
_{2}. A similar space for an equilateral triangle is D_{3}, 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.

**Euclidean space** is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension, including the three-dimensional space and the *Euclidean plane*. It was introduced by the Ancient Greek mathematician Euclid of Alexandria, and the qualifier *Euclidean* is used to distinguish it from other spaces that were later discovered in physics and modern mathematics.

In group theory, the **symmetry group** of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object *X* is *G* = Sym(*X*).

In mathematics, particularly in complex analysis, a **Riemann surface** is a one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In mathematics, **affine geometry** is what remains of Euclidean geometry when not using the metric notions of distance and angle.

In mathematics, the **open unit disk** around *P*, is the set of points whose distance from *P* is less than 1:

In mathematics, an **isometry** is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

In mathematics, a **hyperbolic space** is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature. It is hyperbolic geometry in more than 2 dimensions, and is distinguished from Euclidean spaces with zero curvature that define the Euclidean geometry, and elliptic geometry that have a constant positive curvature.

A tiling or **tessellation** of a flat surface is the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries.

In mathematics, the **conformal group** of a space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.

**Rotation** in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. A rotation is different from other types of motions: translations, which have no fixed points, and (hyperplane) reflections, each of them having an entire (*n* − 1)-dimensional flat of fixed points in a n-dimensional space. A clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude.

In mathematics, a **discrete subgroup** of a topological group *G* is a subgroup *H* such that there is an open cover of *G* in which every open subset contains exactly one element of *H* ; in other words, the subspace topology of *H* in *G* is the discrete topology. For example, the integers, **Z**, form a discrete subgroup of the reals, **R**, but the rational numbers, **Q**, do not. A **discrete group** is a topological group *G* equipped with the discrete topology.

In mathematics, **low-dimensional topology** is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.

In mathematics, a **Euclidean group** is the group of (Euclidean) isometries of an Euclidean space 𝔼^{n}; that is, the transformations of that space that preserve the Euclidean distance between any two points. The group depends only on the dimension *n* of the space, and is commonly denoted E(*n*) or ISO(*n*).

In geometry, **hyperbolic motions** are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and *motion*.

In mathematics, a **hyperbolic metric space** is a metric space satisfying certain metric relations between points. The definition, introduced by Mikhael Gromov, generalizes the metric properties of classical hyperbolic geometry and of trees. Hyperbolicity is a large-scale property, and is very useful to the study of certain infinite groups called (Gromov-)hyperbolic groups.

In geometry, an **apeirogon** or **infinite polygon** is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.

In mathematics—specifically, in differential geometry—a **geodesic map** is a function that "preserves geodesics". More precisely, given two (pseudo-)Riemannian manifolds (*M*, *g*) and (*N*, *h*), a function *φ* : *M* → *N* is said to be a geodesic map if

**Geometry** is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.

In geometry, an object has **symmetry** if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be *symmetric under rotation* or to have *rotational symmetry*. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

- ↑ 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 . - ↑ 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,**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,**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 .

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