# 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.