mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...

differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

and

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

Differential equations

In the theory of

, 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. *Chaplygin inequality * 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 In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after Andrey Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov), KPP equation or Fis ...

, a reaction--diffusion equation. * Hille-Wintner comparison theorem

Riemannian geometry

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

Rauch comparison theorem In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, ...

relates the

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 p ...

of a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...

to the rate at which its

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...

s 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 * Berger–Kazdan comparison theorem * Warner comparison theorem for

length Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the Inte ...

s of N-Jacobi fields (''N'' being a submanifold of a complete Riemannian manifold) *

Bishop–Gromov inequality In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness ...

, conditional on a lower bound for the

Ricci curvature In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

s R.L. Bishop & R. Crittenden, ''Geometry of manifolds'' * Lichnerowicz comparison theorem * Eigenvalue comparison theorem ** Cheng's 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 In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they hav ...

References

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