Journal for Geometry and Graphics 19 (2015), No. 2, 179--188
Copyright Heldermann Verlag 2015
A Property of Liouville Surfaces and Manifolds
Dieselstr. 19, 80993 München, Germany
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].
Keywords: Energy/action of curve, Liouville metric, Liouville surface, Liouville manifold.
MSC: 53A05; 53A07
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