Geodesic maps ``in the large'' of Ricci-flat spaces with \(n\) complete geodesic lines (Q2197261)

The authors study global maps preserving geodesics from a pseudo-Riemannian manifold to Ricci-flat pseudo-Riemannian manifold. Two sufficient conditions for such map to be affine, are found. One of these conditions is the following: The scalar curvature vanishes at a point of the source manifold and complete geodesic lines through this point with complete images pass in \(n\) linearly independent directions.