Geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability

2013/06/28 by Irena Hinterleitner, Hinterleitner, I., Josef Mikeš +1 · 1 citation
Mathematics · Physics and Astronomy · #53B20 #53B21 #53B30 #53C25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · doi:10.48550/arxiv.1306.6810

openalex publication_date 2013/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox(C^r, r≥1). Also, if the Einstein space V_n admits a non trivial geodesic mapping onto a \hbox(pseudo-) Riemannian manifold V_n∈ C¹, then V_n is an Einstein space. If a four-dimensional Einstein space with non constant curvature globally admits a geodesic mapping onto a (pseudo-) Riemannian manifold V₄∈ C¹, then the mapping is affine and, moreover, if the scalar curvature is non vanishing, then the mapping is homothetic, i.e. g=\rm const⋅ g.

Citations

• On geodesic mappings of manifolds with affine connection
• Holomorphically projective mappings and their generalizations

Cited by

• Geodesic Mappings and Their Generalizations

Related

• Geodesic Mappings and Einstein Spaces
• Geodesic Mappings of Special Riemannian Manifolds
• Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces
• On conformal invariance of isotropic geodesics
• Non-reductive Homogeneous Pseudo-Riemannian Manifolds of Dimension Four