Comparison theorem

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In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry.

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.[1][2]

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. [4]

Other[edit]

References[edit]

  1. ^ "Comparison theorem - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
  2. ^ See also: Lyapunov comparison principle
  3. ^ "Differential inequality - Encyclopedia of Mathematics". www.encyclopediaofmath.org. Retrieved 2019-12-13.
  4. ^ Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
  5. ^ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
  6. ^ Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld.
  7. ^ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
  8. ^ R.L. Bishop & R. Crittenden, Geometry of manifolds