Rauch comparison theorem

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. 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.

Statement This section needs attention from an expert in mathematics. See the talk page for details. WikiProject Mathematics may be able to help recruit an expert. (February 2015) Let {displaystyle M,{widetilde {M}}} be Riemannian manifolds, let {displaystyle gamma :[0,T]to M} and {displaystyle {widetilde {gamma }}:[0,T]to {widetilde {M}}} be unit speed geodesic segments such that {displaystyle {widetilde {gamma }}(0)} has no conjugate points along {displaystyle {widetilde {gamma }}} , and let {displaystyle J,{widetilde {J}}} be normal Jacobi fields along {displaystyle gamma } and {displaystyle {widetilde {gamma }}} such that {displaystyle J(0)={widetilde {J}}(0)=0} and {displaystyle |D_{t}J(0)|=left|{widetilde {D}}_{t}{widetilde {J}}(0)right|} . Suppose that the sectional curvatures of {displaystyle M} and {displaystyle {widetilde {M}}} satisfy {displaystyle K(Pi )leq {widetilde {K}}({widetilde {Pi }})} whenever {displaystyle Pi subset T_{gamma (t)}M} is a 2-plane containing {displaystyle {dot {gamma }}(t)} and {displaystyle {widetilde {Pi }}subset T_{{tilde {gamma }}(t)}{widetilde {M}}} is a 2-plane containing {displaystyle {dot {widetilde {gamma }}}(t)} . Then {displaystyle |J(t)|geq |{widetilde {J}}(t)|} for all {displaystyle tin [0,T]} .

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.

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