Differential geometry Curvature and topology Theorem

Toponogov's theorem

Builds on Riemannian manifold Sectional curvature Geodesic Riemannian distance Space form

Statement

Let $(M, g)$ be a connected complete Riemannian manifold of dimension $n \ge 2$ whose sectional curvature satisfies $K \ge k$ for a real number $k$ . Let $M_k^2$ be the simply connected complete surface of constant curvature $k$ , a space form. Let $p \in M$ , and let $\gamma_1, \gamma_2$ be minimizing unit-speed geodesics starting at $p$ , of lengths $\ell_1, \ell_2$ , that make the angle $\alpha \in [0, \pi]$ at $p$ . If $k > 0$ , assume $\ell_1, \ell_2 \le \pi / \sqrt{k}$ . Let $\bar\gamma_1, \bar\gamma_2$ be unit-speed geodesics in $M_k^2$ starting at a point $\bar p$ , of lengths $\ell_1, \ell_2$ , that make the angle $\alpha$ at $\bar p$ . Then $d\bigl(\gamma_1(\ell_1), \gamma_2(\ell_2)\bigr) \le \bar d\bigl(\bar\gamma_1(\ell_1), \bar\gamma_2(\ell_2)\bigr),$ where $d$ is the Riemannian distance of $M$ and $\bar d$ that of $M_k^2$ .

Remarks

This is the hinge form of the theorem. An equivalent form compares the angles of a geodesic triangle with minimizing sides in $M$ with those of the triangle with the same side lengths in $M_k^2$ . Each angle in $M$ is at least the corresponding angle in $M_k^2$ . The proof starts from the Rauch comparison theorem, which gives the result for short hinges, and extends it to all hinges by a subdivision argument that uses completeness.

Example: if $K \ge 0$ , compare with the Euclidean plane. Then for two minimizing geodesics of lengths $\ell_1, \ell_2$ from $p$ at angle $\alpha$ , the endpoints are at distance at most $\sqrt{\ell_1^2 + \ell_2^2 - 2 \ell_1 \ell_2 \cos\alpha}$ , the value given by the law of cosines. On the round sphere, for $0 < \alpha < \pi$ , the distance is strictly smaller, since triangles on a sphere are fatter than flat triangles.

Going deeper

For a student

Toponogov's theorem turns a lower curvature bound into a statement about distances only, without derivatives. This is why it is the basis of many global results such as the soul theorem and the Grove–Shiohama diameter sphere theorem. Its conclusion is also used as the definition of a lower curvature bound for metric spaces, which leads to Alexandrov spaces.

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Tag	`KX0B3AWD5H`
URI	`m2://KX0B3AWD5H`
Kind	statement, theorem
Writing	W0 stub
MSC	53C20, 53C23
Status	This copy of the site has no build data. Run `m2 status KX0B3AWD5H`.
About	Sectional curvature
Uses	Riemannian manifold, Sectional curvature, Geodesic, Riemannian distance, Space form
Same as	Wikidata Q7825061 Wikipedia Toponogov's theorem
Changed	2 November 2026

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