# 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.

### Sturm–Picone comparison theorem

Sturm comparison theoremSturm-Picone comparison theorem

### 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.

### Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds

### Geodesic

geodesicsgeodesic flowgeodesic equation

### Toponogov's theorem

Toponogov theoremToponogov's triangle comparison theoremV. Toponogov

### Myers's theorem

Myers theoremMyers' theoremBonnet–Myers theorem

### Direct comparison test

comparison testLimit comparison theorem

### Spectral sequence

spectral sequencesWang sequencedegeneration

### 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.