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Can someone explain with some simple examples what is meant by the isometry and isometry group? I'm a student of physics and I often come across this term in texts of general relativity for various spaces. In particular, I want to understand the statement why

The isometry group of de Sitter space is the Lorentz group $O(1,n)$.

SRS
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    Did you try the wikipedia articles on isometries and isometry groups? – Arthur May 14 '18 at 15:48
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    An isometry on a (semi-)Riemannian manifold is a diffeomorphism of the manifold into itself so that preserves distances or, equivalently, preserves the Riemannian metric (ie $\phi^* g = g$ where $\phi$ is the diffeomorphism and $g$ the metric). It is elementary that isometries form a group, you can then try to find out what this group is. For example on euclidean $\Bbb R^n$ it is $O(n)\ltimes \Bbb R^n$. For Minkowski-space $\Bbb R^{3,1}$ it is the "Poincaré group" $O(1,3)\ltimes \Bbb R^4$ (actually the Poincaré group is the double cover of this group). – s.harp May 14 '18 at 16:07
  • Normally one would however say that $O(n)$ is the isometry group of $\Bbb R^n$, or that $O(n,1)$ is isometry group of $\Bbb R^{n,1}$. You can recover these groups by looking at the stabiliser (of the action of isometries) of an arbitrary point. – s.harp May 14 '18 at 16:09

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An isometry is a shape preserving transformation. Rotations and reflections are two examples. A dilation is not an isometry because it changes the size of the shape. An example of an isometry group would be all the transformations of a say a regular hexagon (rotations and reflection) that would result in no change in the appearance (symmetry). The group would have 12 elements.

Phil H
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As an example, let $\mathbb{E}$ be Euclidean Space. Then every isometry, $A$ of $\mathbb{E}^N$ can be written as $$A: \bf{x} \mapsto B {\bf x} + \bf{v}$$

With $B\in O(N)$ and $\bf{v} \in \mathbb{E}^N$. Then one can define a map $$\phi: \text{Isom}(\mathbb{E}^N) \rightarrow O(N)$$ By $A \mapsto B$. the group $\text{ Isom}$ is a group homomorphism whose kernel is equal to $\text{Trans}(\mathbb{E}^N)$ of all translations of $\mathbb{E}^N$.