The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered

Suppose all geodesics of two Riemannian metrics g and defined on a (connected, geodesically complete) manifold M ⁿ coincide. At each point x M ⁿ , consider the common eigenvalues ₁, ₂, ... , _n of the two metrics (we assume that _{1 2 n}) and the numbers . We show that the numbers _i are ordered over the entire manifold: for any two points x and y in M the number _k(x) is not greater than _k+1(y). If _k(x)= _k+1(y), then there is a point z M _n such that _k(z)= _k+1(z). If the manifold is closed and all the common eigenvalues of the metrics are pairwise distinct at each point, then the manifold can be covered by the torus.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 412–423.Original Russian Text Copyright © 2005 by V. S. Matveev.This revised version was published online in April 2005 with a corrected issue number.