The Rauch Comparison Theorem, named for Harry Rauch, who proved it in 1951, is a fundamental result of Riemannian geometry relating the sectional curvature of a manifold to the rate at which nearby geodesics spread apart or converge. It states, roughly, that where curvature is larger, geodesics emanating from a point tend to converge more quickly, while where curvature is smaller, or negative, they tend to spread apart more quickly, giving a precise comparison between a manifold's own geodesics and those of a simpler model space of constant curvature. The theorem is a basic tool underlying many later global comparison results in Riemannian geometry.
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StatementFor positive curvature geodesics tend to converge, while for negative curvature geodesics tend to spread; comparing sectional curvatures bounds the growth of Jacobi fields accordingly. 1 Classification
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Source Rauch comparison theorem, Wikipedia
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proved it in 1951
Intuitive summary
for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread
Proved By: Harry Rauch, Lead sentence
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.
- In Branch: Differential Geometry, Lead sentence
View the SourceHarry Rauch (Wikipedia)
Principal Result Of: Harry Rauch, Body, workQuote, Principal Result Of: Harry Rauch, Body, work
The Rauch comparison theorem is also named after Harry Rauch. He proved it in 1951.
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