# 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
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.

### 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.