In the

mathematical Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

field of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

, Toponogov's theorem (named after Victor Andreevich Toponogov) 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 is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

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 po ...

''K'' satisfying

K\ge \delta\,.

Let ''pqr'' be a geodesic triangle, 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 (topology), path between two points can be continuously transformed into any other such path while preserving ...

space of constant curvature 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 yields an analogous statement, but with the reverse inequality .

References

* * *

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 {{Riemannian-geometry-stub