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

${\displaystyle z=f(x,y)}$

such that the first fundamental form is of the form

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

Sometimes a metric of this form is called a Liouville metric. Every surface of revolution is a Liouville surface.

References

• 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.