Comparison theorem

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In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, [1] and often occur in fields such as calculus, differential equations and Riemannian geometry.


Differential equations

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. [2] [3]

Riemannian geometry

In Riemannian geometry, it is a traditional name for a number of theorems that compare various metrics and provide various estimates in Riemannian geometry. [5]


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In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. In particular it implies that every Riemann surface admits a Riemannian metric of constant curvature. For compact Riemann surfaces, those with universal cover the unit disk are precisely the hyperbolic surfaces of genus greater than 1, all with non-abelian fundamental group; those with universal cover the complex plane are the Riemann surfaces of genus 1, namely the complex tori or elliptic curves with fundamental group Z2; and those with universal cover the Riemann sphere are those of genus zero, namely the Riemann sphere itself, with trivial fundamental group.

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  1. "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-12-13.
  2. "Comparison theorem - Encyclopedia of Mathematics". Retrieved 2019-12-13.
  3. See also: Lyapunov comparison principle
  4. "Differential inequality - Encyclopedia of Mathematics". Retrieved 2019-12-13.
  5. Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
  6. M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700712
  7. Weisstein, Eric W. "Berger-Kazdan Comparison Theorem". MathWorld .
  8. F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341356
  9. R.L. Bishop & R. Crittenden, Geometry of manifolds