Proceedings of the International Conference on Geometry, Integrability and Quantization

Geodesic Mappings Onto Riemannian Manifolds and Differentiability

Abstract

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto (pseudo-)Riemannian manifolds. We proved that if a manifold with affine (or projective) connection of differentiability class $C^r (r\geq2)$ admits a geodesic mapping onto a (pseudo-)Riemannian manifold of class $C^1$, then this manifold belongs to the differentiability class $C^{r+1}$. From this result follows if an Einstein spaces admits non-trivial geodesic mappings onto (pseudo-)Riemannian manifolds of class $C^1$ then this manifold is an Einstein space, and there exists a common coordinate system in which the components of the metric of these Einstein manifolds are real analytic functions.

Article information

Dates
First available in Project Euclid: 14 January 2017

https://projecteuclid.org/ euclid.pgiq/1484362823

Digital Object Identifier
doi:10.7546/giq-18-2017-183-190

Mathematical Reviews number (MathSciNet)
MR3616920

Zentralblatt MATH identifier
1378.53025

Citation

Hinterleitner, Irena; Mikeš, Josef. Geodesic Mappings Onto Riemannian Manifolds and Differentiability. Proceedings of the Eighteenth International Conference on Geometry, Integrability and Quantization, 183--190, Avangard Prima, Sofia, Bulgaria, 2017. doi:10.7546/giq-18-2017-183-190. https://projecteuclid.org/euclid.pgiq/1484362823