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 R<sup>3</sup>
such that the first fundamental form is of the form
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 with metric
where and are functions of and and are functions of . A geodesic line on such a surface is given by
and the distance along the geodesic is given by
Here is a constant related to the direction of the geodesic by
where is the angle of the geodesic measured from a line of constant . 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.