Rauch Comparison Theorem

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.