In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

, Toponogov's theorem (named after

Victor Andreevich Toponogov Victor Andreevich Toponogov (russian: Ви́ктор Андре́евич Топоно́гов; March 6, 1930 – November 21, 2004) was an outstanding Russian mathematician, noted for his contributions to differential geometry and so-called ...

) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics emanating from a point ''p'' spread apart more slowly in a region of high curvature than they would in a region of low curvature. Let ''M'' be an ''m''-dimensional

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

with

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

''K'' satisfying

K\ge \delta\,.

Let ''pqr'' be a

geodesic triangle In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

, i.e. a triangle whose sides are geodesics, in ''M'', such that the geodesic ''pq'' is minimal and if δ > ''0'', the length of the side ''pr'' is less than

\pi / \sqrt \delta

. Let ''p''′''q''′''r''′ be a geodesic triangle in the model space ''M''_δ, i.e. the

simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...

space of

constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...

δ, such that the lengths of sides ''p′q′'' and ''p′r′'' are equal to that of ''pq'' and ''pr'' respectively and the angle at ''p′'' is equal to that at ''p''. Then :

d(q,r) \le d(q',r').\,

When the sectional curvature is bounded from above, a corollary to the

Rauch comparison theorem In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, ...

yields an analogous statement, but with the reverse inequality .

References

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External links

Pambuccian V., Zamfirescu T. "Paolo Pizzetti: The forgotten originator of triangle comparison geometry". ''Hist Math'' 38:8 (2011)
Theorems in Riemannian geometry Geometric inequalities {{differential-geometry-stub