Comparison theorem

comparecomparison methodcomparison theorem in Riemannian geometry
A comparison theorem is any of a variety of theorems that compare properties of various mathematical objects.wikipedia
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Grönwall's inequality

Grönwall's lemmaGrönwall–Bellman inequality
In particular, it provides a comparison theorem that can be used to prove uniqueness of a solution to the initial value problem; see the Picard–Lindelöf theorem.

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
If tighter bounds on the sectional curvature are known, then this property generalizes to give a comparison theorem between geodesic triangles in M and those in a suitable simply connected space form; see Toponogov's theorem.

Bishop–Gromov inequality

In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov.

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
comparison theorem) with the geometry of a constant curvature space form.

Differential equation

differential equationsdifferentialsecond-order differential equation
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.

Fisher's equation

Fisher–Kolmogorov equationKolmogorov–Petrovsky–Piskunov equationFisher-Kolmogorov equation

Riemannian geometry

Riemannianlocal to global theoremsRiemann 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.

Roberto Conti (mathematician)

Roberto ContiConti, Roberto
Roberto Conti (23 April 1923 – 30 August 2006) was an Italian mathematician, who contributed to the theory of ordinary differential equations and the development of the comparison method.

Positive energy theorem

positive mass conjecturepositive mass theorem
The theorem is a scalar curvature comparison theorem, with asymptotic boundary conditions, and a corresponding statement of geometric rigidity.

Grigori Perelman

PerelmanG. PerelmanPerelman, Grigori
Until late 2002, Perelman was best known for his work in comparison theorems in Riemannian geometry.

Curvature

curvednegative curvatureextrinsic curvature
Another generalization of curvature relies on the ability to compare a curved space with another space that has constant curvature.