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

$d{s}^2=(f_1(x)+f_2(y))(d{x}^2+d{y}^2).$

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

$d{s}^2=(U-V)(U_1^{2}d{u}^2+V_1^{2}d{v}^2),$

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

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

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) (in fr). Leçons sur la théorie générale des surfaces. 3. Gauthier-Villars. https://gallica.bnf.fr/ark:/12148/bpt6k778307/f9. 
• 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. 
• "Note von der geodätischen Linie auf einem Ellipsoid und den verschiedenen Anwendungen einer merkwürdigen analytischen Substitution" (in de). Journal für die Reine und Angewandte Mathematik 1839 (19): 309–313. 1839. doi:10.1515/crll.1839.19.309. https://books.google.com/books?id=RbwGAAAAYAAJ&pg=PA309. 
• Liouville, Joseph (1846). "Sur quelques cas particuliers où les équations du mouvement d'un point matériel peuvent s'intégrer" (in fr). Journal de Mathématiques Pures et Appliquées 11: 345-378. http://sites.mathdoc.fr/JMPA/PDF/JMPA_1846_1_11_A45_0.pdf.