Geodesic
A Property of Liouville Surfaces and Manifolds
Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 179-188.

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We use the definition of the energy of a curve on a surface and show that in a Liouville surface the energy integrals along the diagonals of a net rectangle (Liouville maps are conformal) are equal. This result allows a generalization to Liouville manifolds, which is stated and proved in this paper. A series of different surfaces with an induced Liouville metric are given in Euclidean spaces. One example is given in the pseudo-Euclidean (Minkowski) plane. The material presented here also relates to our previous article in this journal [Journal of Geometry and Graphics 18 (2014) 7--21].
Classification : 53A05, 53A07
Mots-clés : Energy/action of curve, Liouville metric, Liouville surface, Liouville manifold 
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     title = {A {Property} of {Liouville} {Surfaces} and {Manifolds}},
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C.-S. Barbat . A Property of Liouville Surfaces and Manifolds. Journal for geometry and graphics, Tome 19 (2015) no. 2, pp. 179-188. http://geodesic.mathdoc.fr/item/JGG_2015_19_2_JGG_2015_19_2_a1/