Liouville surface

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

$z=f(x,y)$

such that the first fundamental form is of the form

$ds^{2}={\big (}f_{1}(x)+f_{2}(y){\big )}\left(dx^{2}+dy^{2}\right).$

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 $(u,v)$ with metric

$ds^{2}=(U-V)(U_{1}^{2}\,du^{2}+V_{1}^{2}\,dv^{2}),$

where $U$ and $U_{1}$ are functions of $u$ and $V$ and $V_{1}$ are functions of $v$. A geodesic line on such a surface is given by

${\frac {U_{1}\,du}{\sqrt {U-\alpha }}}-{\frac {V_{1}\,dv}{\sqrt {\alpha -V}}}=0$

and the distance along the geodesic is given by

$ds={\frac {UU_{1}\,du}{\sqrt {U-\alpha }}}-{\frac {VV_{1}\,dv}{\sqrt {\alpha -V}}}.$

Here $\alpha$ is a constant related to the direction of the geodesic by

$\alpha =U\sin ^{2}\omega +V\cos ^{2}\omega ,$

where $\omega$ is the angle of the geodesic measured from a line of constant $v$. 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.

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.