# 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.wikipedia

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### Harry Rauch

**Rauch, Harry**

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.

The Rauch comparison theorem is also named after Harry Rauch.

### Jacobi field

**Jacobi equation**

This theorem is formulated using Jacobi fields to measure the variation in geodesics. Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along, and let be normal Jacobi fields along \gamma and such that and.

### Toponogov's theorem

**Toponogov theoremToponogov's triangle comparison theoremV. Toponogov**

*Toponogov's theorem

When the sectional curvature is bounded from above, a corollary to the Rauch comparison theorem yields an analogous statement, but with the reverse inequality.

### Riemannian geometry

**Riemannianlocal to global theoremsRiemann geometry**

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.

### Sectional curvature

**curvaturecurvature tensorsmanifolds with constant sectional curvature**

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.

### Riemannian manifold

**Riemannian metricRiemannianRiemannian manifolds**

### Geodesic

**geodesicsgeodesic flowgeodesic equation**

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. Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along, and let be normal Jacobi fields along \gamma and such that and.

### Conjugate points

**conjugateConjugate point**

Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along, and let be normal Jacobi fields along \gamma and such that and.

### Manfredo do Carmo

**M. do CarmoManfredo P. do CarmoManfredo Perdigão do Carmo**

### Comparison theorem

**comparecomparison methodcomparison theorem in Riemannian geometry**

### Foundations of Differential Geometry

It continues with geodesics on Riemannian manifolds, Jacobi fields, the Morse index, the Rauch comparison theorems, and the Cartan–Hadamard theorem.

### CR manifold

**CR geometryCR structurecomplex-real**

These spaces can be used as comparison spaces in studying geodesics and volume comparison theorems on CR manifolds with zero Webster torsion akin to the H.E. Rauch comparison theorem in Riemannian Geometry.