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

Formella S.

Opuscula Mathematica

2005

Vol. 25, no. 2

181-187

Let M be a differentiable manifold and denote by nabla and nabla~ two linear connections on M. Nabla and 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 nabla~ = nabla - T are geodesically equivalent, where T is a homogeneous structure on (M, g) ([7]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold (M, g) onto (M, nabla~), then the map is affine. If a naturally reductive manifold (M, g) admits a nontrivial geodesic mapping onto a Riemannian manifold (formula) then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings (M, g) arr (M, nabla~).