A note on geodesic and almost geodesic mappings of homogeneous Riemannian manifolds

Abstract

Let $M$ be a differentiable manifold and denote by $\nabla$ and $\tilde{\nabla}$ two linear connections on $M$. $\nabla$ and $\tilde{\nabla}$ are said to be geodesically equivalent if and only if they have the same geodesics. A Riemannian manifold $(M,g)$ is a naturally reductive homogeneous manifold if and only if $\nabla$ and $\tilde{\nabla}=\nabla-T$ are geodesically equivalent, where $T$ is a homogeneous structure on $(M,g)$ ([Tricerri F., Vanhecke L., Homogeneous Structure on Riemannian Manifolds. London Math. Soc. Lecture Note Series, vol. 83, Cambridge Univ. Press 1983]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold $(M,g)$ onto $(M,\tilde{\nabla})$, then the map is affine. If a naturally reductive manifold $(M,g)$ admits a nontrivial geodesic mapping onto a Riemannian manifold $(\overline{M},\overline{g})$ then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings $(M,g) \to (M,\tilde{\nabla})$.