Comparison theorem

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A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.

Differential equations[edit]

In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof) provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. See also Lyapunov comparison principle

Riemannian geometry[edit]

In Riemannian geometry it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry.

See also: Comparison triangle



  1. ^ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
  2. ^ Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld.
  3. ^ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
  4. ^ R.L. Bishop & R. Crittenden, Geometry of manifolds