an:03262928
Zbl 0164.22103
Robinson, J.; Zund, J. D.
A theorem on geodesic mappings
EN
Tensor, New Ser. 19, 300-302 (1968).
00135758
1968
j
53-XX
differential geometry
This paper proves that there does not exist a non-trivial geodesic mapping
\[ \Gamma_{ij}^h \to \hat{\Gamma}_{ij}^h = \Gamma_{ij}^h+\delta_i^h\psi_j+\delta_j^h\psi_j,\quad \psi_i\ne 0 \]
which takes a Riemannian \(V_n\) \((n>2)\) onto a \(\tilde V_n\), which is recurrent, but is not of constant curvature. Thus the celebrated Beltrami theorem of classical differential geometry does not admit nontrivial generalizations. The paper corrects and extends a previous result of \textit{N. S. Sinyukov} [Dokl. Akad. Nauk SSSR, n. Ser. 98, 21--23 (1954; Zbl 0056.15301)].
J. Robinson
Zbl 0056.15301