Liouville surface

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2020 Mathematics Subject Classification: Primary: 53A05 [MSN][ZBL]

A surface for which the equations of the geodesics admit a quadratic integral $a_{ij} du^i du^j$, where the tensor $a_{ij}$ is distinct from the metric tensor $g_{ij}$ 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.


[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)
How to Cite This Entry:
Liouville surface. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article