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.

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.