Geodesic transformations and harmonic spaces (Q1284648)

The authors define a geodesic transformation \(\phi_P\) with respect to the point \(P\) in a Riemannian manifold (this transformation is not a geodesic transformation in the classical sense of T. Fubini). Such transformations of harmonic spaces are studied. The authors prove that a Riemannian manifold is harmonic if and only if there exists a nontrivial divergence-preserving geodesic transformation (\(\phi_P\)) with respect to each point which is not a geodesic symmetry.