# 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

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]}

- Chaplygin inequality
^{[3]} - Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations.
- Sturm comparison theorem
- Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation.
- Hille-Wintner comparison theorem

## 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. ^{[4]}

- Rauch comparison theorem relates the sectional curvature of a Riemannian manifold to the rate at which its geodesics spread apart.
- Toponogov's theorem
- Myers's theorem
- Hessian comparison theorem
- Laplacian comparison theorem
- Morse–Schoenberg comparison theorem
- Berger comparison theorem, Rauch–Berger comparison theorem
^{[5]} - Berger–Kazdan comparison theorem
^{[6]} - Warner comparison theorem for lengths of N-Jacobi fields (
*N*being a submanifold of a complete Riemannian manifold)^{[7]} - Bishop–Gromov inequality, conditional on a lower bound for the Ricci curvatures
^{[8]} - Lichnerowicz comparison theorem
- Eigenvalue comparison theorem
- See also: Comparison triangle

## Other

- Limit comparison theorem, about convergence of series
- Comparison theorem for integrals, about convergence of integrals
- Zeeman's comparison theorem, a technical tool from the theory of spectral sequences

## References

- ↑ "Comparison theorem - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Comparison_theorem.
- ↑ See also: Lyapunov comparison principle
- ↑ "Differential inequality - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Differential_inequality.
- ↑ Jeff Cheeger and David Gregory Ebin: Comparison theorems in Riemannian Geometry, North Holland 1975.
- ↑ M. Berger, "An Extension of Rauch's Metric Comparison Theorem and some Applications", Illinois J. Math., vol. 6 (1962) 700–712
- ↑ Weisstein, Eric W.. "Berger-Kazdan Comparison Theorem". http://mathworld.wolfram.com/Berger-KazdanComparisonTheorem.html.
- ↑ F.W. Warner, "Extensions of the Rauch Comparison Theorem to Submanifolds" (Trans. Amer. Math. Soc., vol. 122, 1966, pp. 341–356
- ↑ R.L. Bishop & R. Crittenden,
*Geometry of manifolds*

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