On Riemannian manifolds with topological transitive geodesic flows (Q1175972)

The author obtains some properties of complete Riemannian manifolds \((M,g)\) with topologically transitive geodesic flow. The most interesting of them are the following: (i) The Lie group of isometries of \((M,g)\) is discrete. (ii) Let \(f: (M,g)\rightarrow (M',g')\) be a diffeomorphism that maps non-parametrized geodesics to non- parametrized geodesics. Then \(f\) is a homothety. (iii) Let \(\rho\) be the Ricci tensor of \((M,g)\). If \(\nabla_ x\rho(X,X)=0\) for every vector field \(X\) on \(M\), then \(\rho=\text{const.}g\).