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created: 2023-03-26T11:58
updated: 2024-05-03T23:22
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# constant curvature

## [Dolgachev, REFLECTION GROUPS IN ALGEBRAIC GEOMETRY](https://dept.math.lsa.umich.edu/~idolga/reflections.pdf)

A space of constant curvature is a simply connected Riemannian homogeneous space $X$ such that the isotropy subgroup of its group of isometries $\operatorname{Iso}(X)$ at each point coincides with the full orthogonal group of the tangent space. Up to isometry and rescaling the metric, there are three spaces of constant curvature of fixed dimension $n$.
- The euclidean space $E^n$ with $\operatorname{Iso}(X)$ equal to the affine orthogonal group $\mathrm{AO}^n=\mathbb{R}^n \rtimes \mathrm{O}(n)$.
- The $n$-dimensional sphere
$$
S^n=\left\{\left(x_0, \ldots, x_n\right) \in \mathbb{R}^{n+1}: x_0^2+\ldots+x_n^2=1\right\}
$$
with $\operatorname{Iso}(X)$ equal to the orthogonal group $\mathrm{O}(n+1)$.
- The hyperbolic (or Lobachevsky) space
$$
H^n=\left\{\left(x_0, \ldots, x_n\right) \in \mathbb{R}^{n+1},-x_0^2+x_1^2+\ldots+x_n^2=-1, x_0>0\right\}
$$
with $\operatorname{Iso}(X)$ equal to the subgroup $\mathrm{O}(n, 1)^{+}$of index 2 of the orthogonal group $\mathrm{O}(n, 1)$ which consists of transformations of spinor norm 1 , that is, transformations that can be written as a product of reflections in vectors with positive norm. The Riemannian metric is induced by the hyperbolic metric in $\mathbb{R}^{n+1}$,
$$
d s^2=-d x_0^2+d x_1^2+\ldots+d x_n^2
$$