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arXiv · 1306.6810

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

I. Hinterleitner·J. Mikeš
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Abstract

In this paper we prove that geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability \hbox{$(C^r, r\geq1)$}. Also, if the Einstein space $V_n$ admits a non trivial geodesic mapping onto a \hbox{(pseudo-)} Riemannian manifold $\bar V_n\in C^1$, then $\bar V_n$ is an Einstein space. If a four-dimensional Einstein space with non constant curvature globally admits a geodesic mapping onto a (pseudo-) Riemannian manifold $\bar V_4\in C^1$, then the mapping is affine and, moreover, if the scalar curvature is non vanishing, then the mapping is homothetic, i.e. $\bar g={\rm const}\cdot g$.

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I. Hinterleitner, J. Mikeš. 2013-06-28. Geodesic mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability. https://arxiv.org/abs/1306.6810

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