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The author previously studied with {\it F. Ilosvay} and {\it B. Kis} [Publ. Math. 42, 139-144 (1993; Zbl 0796.53022)] the diffeomorphisms between two Finsler spaces $F^n= (M^n, L)$ and $\overline F^n= (M^n,\overline L)$ which map the geodesics of $F^n$ to geodesics of $\overline F^n$ (geodesic mappings).\par Now, he investigates the geodesic mappings between a Finsler space $F^n$ and a Riemannian space $\overline{\bbfR}^n$. The main result of this paper is as follows: if $F^n$ is of constant curvature $K$ and the mapping $F^n\to \overline{\bbfR}^n$ is a strongly geodesic mapping then $K= 0$ or $K\ne 0$ and $\overline L= e^{\varphi(x)}L$.
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