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. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread.

The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with constant curvature, and the second one is the manifold under study : a bound (either lower or upper) on its sectional curvature is then needed in order to apply Rauch comparison theorem.

Statement

be Riemannian manifolds, on which are drawn unit speed geodesic has no conjugate points be two normal Jacobi fields such that :

• .

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Conditions of the theorem

The theorem is formulated using Jacobi fields. Up to reparametrization, every variation of geodesics induces a normal Jacobi field.

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Analog theorems

Under very similar conditions, it is also possible to compare the Hessian of the distance function to a given point. It is also possible to compare the Laplacian of this function (which is the trace of the Hessian), with some additional condition on one of the two manifolds: it is then enough to have an inequality on the Ricci curvature (which is the trace of the curvature tensor).

See also

• Toponogov's theorem

References

• do Carmo, M.P. Riemannian Geometry, Birkhäuser, 1992.

• Lee, J. M., Riemannian Manifolds: An Introduction to Curvature, Springer, 1997.