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A theorem on geodesic mappings. (English) Zbl 0164.22103
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 N. S. Sinyukov [Dokl. Akad. Nauk SSSR, n. Ser. 98, 21–23 (1954; Zbl 0056.15301)].
Reviewer: J. Robinson

##### MSC:
 53-XX Differential geometry
##### Keywords:
differential geometry