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

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• Template:Cite book (Translated from the Russian by R. Silverman.)
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