Beltrami's theorem

In mathematics — specifically, in Riemannian geometry — Beltrami's theorem is a result named after the Italian mathematician Eugenio Beltrami which states that geodesic maps preserve the property of having constant curvature. More precisely, if (M, g) and (N, h) are two Riemannian manifolds and φ : M → N is a geodesic map between them, and if either of the manifolds (M, g) or (N, h) has constant curvature, then so does the other one.

References

• Ambartzumian, R. V. (1982). Combinatorial integral geometry. Wiley Series in Probability and Mathematical Statistics: Tracts on Probability and Statistics. New York: John Wiley & Sons Inc. p. 26. ISBN 0-471-27977-3. MR 679133. 
• Kreyszig, Erwin (1991). Differential geometry. New York: Dover Publications Inc. ISBN 0-486-66721-9. MR 1118149. 

External links

• Weisstein, Eric W., "Beltrami's theorem", MathWorld.