Boris Kruglikov (University of Tromsoe)

Strictly non-proportional geodesically equivalent metrics have $h_$top$(g)=0$

Abstract. This is a joint result with Vladimir Matveev.

Let $g$ be a Riemannian metric on a closed manifold $M$ and $f:M\to M$ be a projective transformation, i.e. a diffeomorphism preserving non-parametrized geodesics. Assume that it is non-degenerate in a sense that the endomorphisms field $g^{-1}f^*g$ of $TM$ has simple spectrum at some point of $M$ (i.e. metrics $g$ and $f^*g$ are strictly-non-proportional at this point). Then the topological entropy of the geodesic flow of $g$ vanishes.

As a corollary we conclude that a rationally hyperbolic closed manifold does not admit two geodesically equivalent Riemannian metrics, which are strictly non-proportional somewhere. Morover, in the non-simply connected case a manifold admitting such a pair of metrics can be covered by the product of a rationally-elliptic manifold and a torus.

The obtained results are sharp: there are geodesic flows with $h_$top$(g)>0$ admitting (degenerate in the above sense) projective transformations and many examples of rationally elliptic manifolds with non-trivially geodesically equivalent metrics are known.