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 Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, differential equations 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 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. *Chaplygin inequality *

Grönwall's inequality In mathematics, Grönwall's inequality (also called Grönwall's lemma or the Grönwall–Bellman inequality) allows one to bound a function that is known to satisfy a certain differential inequality, differential or integral inequality by the soluti ...

, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations. *

Sturm comparison theorem Sturm (German for storm) may refer to: People * Sturm (surname), surname (includes a list) * Saint Sturm (died 779), 8th-century monk Food * Federweisser, known as ''Sturm'' in Austria, wine in the fermentation stage * Sturm Foods, an American d ...

*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. Intuitivel ...

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 poi ...

of a Riemannian manifold to the rate at which its geodesics spread apart. *

Toponogov's theorem In the mathematical field of Riemannian geometry, Toponogov's theorem (named after Victor Andreevich Toponogov) is a triangle comparison theorem. It is one of a family of comparison theorems that quantify the assertion that a pair of geodesics ema ...

Myers's theorem Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following: In the special case of ...

* Hessian comparison theorem * Laplacian comparison theorem * Morse–Schoenberg comparison theorem *

Berger comparison theorem Berger is a surname in both German and French, although there is no etymological connection between the names in the two languages. The French surname is an occupational name for a shepherd, from Old French ''bergier'' (Late Latin ''berbicarius'', ...

, Rauch–Berger comparison theorem *

Berger–Kazdan comparison theorem In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the -dimens ...

Warner comparison theorem Warner can refer to: People * Warner (writer) * Warner (given name) * Warner (surname) Fictional characters * Yakko, Wakko, and Dot Warner, stars of the animated television series ''Animaniacs'' * Aaron Warner, a character in '' Shatter Me ...

for lengths 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 Define M_^ as the 2-dimensional metric space of constant curvature k. So, for example, M_^ is the Euclidean plane, M_^ is the surface of the unit sphere, and M_^ is the hyperbolic plane. Let X be a metric space. Let T be a triangle in X, with vert ...

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

{{sia, mathematics Mathematical theorems