Let 
be a geodesic mapping. If either 
or 
has constant curvature, then both surfaces have constant curvature
(Ambartzumian 1982, p. 26; Kreyszig 1991).
Beltrami's Theorem
See also
Geodesic MappingExplore with Wolfram|Alpha
References
Ambartzumian, R. V. Combinatorial Integral Geometry. Chichester, England: Wiley, 1982.Kreyszig, E. §91 in Differential Geometry. New York: Dover, 1991.Referenced on Wolfram|Alpha
Beltrami's TheoremCite this as:
Weisstein, Eric W. "Beltrami's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BeltramisTheorem.html