Liouville surface

A surface for which the equations of the geodesics admit a quadratic integral , where the tensor is different from the metric tensor of the surface. For example, a surface of constant Gaussian curvature is a Liouville surface. For a surface to admit a geodesic mapping onto a plane it is necessary and sufficient that it be a Liouville surface (Dini's theorem). See also Liouville net.

References

[a1]	W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
[a2]	M. Berger, B. Gostiaux, "Differential geometry: manifolds, curves, and surfaces" , Springer (1988) (Translated from French)