Sunday, 22 April 2007

dg.differential geometry - Lipschitz equivalence of Riemannian metrics

As Dmitri says any two Riemannian metrics on a compact manifold are Lipschitz equivalent. The proof is quite simple.



Consider g and h two metrics on M, Let UM be the unit tangent bundle, since M is compact, UM is compact. Then you see that f:UMtomathbbR defined by f(x)=fracg(x,x)h(x,x) is continuous and strictly positive. By compactness, it is bounded above and below by positive constants.

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