In the mathematical field of differential geometry a Liouville surface (named after Joseph Liouville) is a type of surface which in local coordinates may be written as a graph in R3
z = f ( x , y ) {\displaystyle z=f(x,y)}
such that the first fundamental form is of the form
d s 2 = ( f 1 ( x ) + f 2 ( y ) ) ( d x 2 + d y 2 ) . {\displaystyle ds^{2}={\big (}f_{1}(x)+f_{2}(y){\big )}\left(dx^{2}+dy^{2}\right).}
Sometimes a metric of this form is called a Liouville metric. Every surface of revolution is a Liouville surface. Darboux gives a general treatment of such surfaces considering a two-dimensional space ( u , v ) {\displaystyle (u,v)} with metric
d s 2 = ( U − V ) ( U 1 2 d u 2 + V 1 2 d v 2 ) , {\displaystyle ds^{2}=(U-V)(U_{1}^{2}\,du^{2}+V_{1}^{2}\,dv^{2}),}
where U {\displaystyle U} and U 1 {\displaystyle U_{1}} are functions of u {\displaystyle u} and
V {\displaystyle V} and V 1 {\displaystyle V_{1}} are functions of v {\displaystyle v} . A geodesic line on such a surface is given by
U 1 d u U − α − V 1 d v α − V = 0 {\displaystyle {\frac {U_{1}\,du}{\sqrt {U-\alpha }}}-{\frac {V_{1}\,dv}{\sqrt {\alpha -V}}}=0}
and the distance along the geodesic is given by
d s = U U 1 d u U − α − V V 1 d v α − V . {\displaystyle ds={\frac {UU_{1}\,du}{\sqrt {U-\alpha }}}-{\frac {VV_{1}\,dv}{\sqrt {\alpha -V}}}.}
Here α {\displaystyle \alpha } is a constant related to the direction of the geodesic by
α = U sin 2 ω + V cos 2 ω , {\displaystyle \alpha =U\sin ^{2}\omega +V\cos ^{2}\omega ,}
where ω {\displaystyle \omega } is the angle of the geodesic measured from a line of constant v {\displaystyle v} . In this way, the solution of geodesics on Liouville surfaces is reduced to quadrature. This was first demonstrated by Jacobi for the case of geodesics on a triaxial ellipsoid, a special case of a Liouville surface.
Notes
References Darboux, Jean-Gaston (1894). Leçons sur la théorie générale des surfaces [Lessons on the General Theory of Surfaces] (in French). Vol. 3. Gauthier-Villars. Gelfand, I.M. & Fomin, S.V. (2000). Calculus of variations. Dover. ISBN 0-486-41448-5. (Translated from the Russian by R. Silverman.) Guggenheimer, Heinrich (1977). "Chapter 11: Inner geometry of surfaces". Differential Geometry. Dover. ISBN 0-486-63433-7. Jacobi, C. G. J. (1839). "Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution" [The geodesic on an ellipsoid and various applications of a remarkable analytical substitution]. Journal für die Reine und Angewandte Mathematik (in German). 1839 (19): 309–313. doi:10.1515/crll.1839.19.309. S2CID 121670851. Liouville, Joseph (1846). "Sur quelques cas particuliers où les équations du mouvement d'un point matériel peuvent s'intégrer" [Special cases where the equations of motion are integrable] (PDF). Journal de Mathématiques Pures et Appliquées (in French). 11: 345–378.
