Relates sectional curvature of a Riemannian manifold to the rate geodesics spread apart
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. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. This theorem is formulated using
Jacobi fields to measure the variation in geodesics.
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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 and such that and . Suppose that the sectional curvatures of and satisfy whenever is a 2-plane containing and is a 2-plane containing . Then for all .