Space form

Builds on Riemannian manifold Sectional curvature Riemannian distance

Definition

Let $n \ge 2$ and let $K$ be a real number. A space form of curvature $K$ is a Riemannian manifold $(M, g)$ of dimension $n$ such that:

$M$ is connected.
$M$ is complete as a metric space for the Riemannian distance.
The sectional curvature of every 2-plane at every point of $M$ equals $K$ .

Remarks

Example: the flat torus $\R^2 / \Z^2$ with the metric induced from $\R^2$ is connected and compact, hence complete, and its sectional curvature is $0$ everywhere. So it is a space form of curvature $0$ .

Non-example: the open unit disc in $\R^2$ with the Euclidean metric is connected and flat, but the Cauchy sequence $(1 - 1/k, 0)$ has no limit in the disc. It fails clause 2.

The simply connected space forms are the sphere of radius $1/\sqrt{K}$ for $K > 0$ , Euclidean space $\R^n$ for $K = 0$ , and hyperbolic space of curvature $K$ for $K < 0$ . Some authors call only these three models space forms.

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Tag	`TBBDP94GDN`
URI	`m2://TBBDP94GDN`
Kind	definition
Writing	W0 stub
MSC	53C20, 53C21
Defines	Space form
Uses	Riemannian manifold, Sectional curvature, Riemannian distance
Changed	2 November 2026

Space form

Definition

Remarks

Reading path

Further reading

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