Space form

In mathematics, a space form is a complete Riemannian manifold M of constant sectional curvature K. The three obvious examples are Euclidean n-space, the n-dimensional sphere, and hyperbolic space, although a space form need not be simply connected.

The Killing–Hopf theorem of Riemannian geometry states that the universal cover of an n-dimensional space form ${\displaystyle M^{n}}$ with curvature ${\displaystyle K=-1}$ is isometric to ${\displaystyle H^{n}}$, hyperbolic space, with curvature ${\displaystyle K=0}$ is isometric to ${\displaystyle R^{n}}$, Euclidean n-space, and with curvature ${\displaystyle K=+1}$ is isometric to ${\displaystyle S^{n}}$, the n-dimensional sphere of points distance 1 from the origin in ${\displaystyle R^{n+1}}$.

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