For each $n \ge 2$ and $k \in \mathbb R$, there is, up to isometry, a unique simply connected complete $n$-dimensional space form of constant sectional curvature $k$. It is $\mathbb R^n$ when $k=0$, the round sphere of radius
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\begin{align*}
\frac{1}{\sqrt{k}}
\end{align*}
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when $k>0$, and hyperbolic space with curvature $k$ when $k<0$.