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

Abstract

Suppose all geodesics of two Riemannian metrics g and $\bar g$ defined on a (connected, geodesically complete) manifold M ⁿ coincide. At each point x ∈ M ⁿ, consider the common eigenvalues ρ ₁, ρ₂, ... , ρ_n of the two metrics (we assume that ρ₁ ≥ ρ₂ ≥ ⋯ ρ_n) and the numbers $\lambda _i = \left( {\rho _1 \rho _2 \cdot \cdot \cdot \rho _n } \right)^{{1 \mathord{\left/ {\vphantom {1 {\left( {n + 1} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {n + 1} \right)}}} \frac{1}{{\rho _i }}$. We show that the numbers λ_i are ordered over the entire manifold: for any two points x and y in M the number λ_k(x) is not greater than λ _k+1(y). If λ_k(x)=λ _k+1(y), then there is a point z ∈ M _n such that λ_k(z)=λ _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.