Liouville surface explained

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

ds ^{2} = ( f _{1(x)} + f ^{2+ dy} _{2 (y) ) \left(dx} ^{2\right).}

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

References

• Book: Gelfand, I.M. . Fomin, S.V. . amp. Calculus of variations. 2000. Dover. 0-486-41448-5. (Translated from the Russian by R. Silverman.)
• Book: Guggenheimer, Heinrich. Differential Geometry. 1977. Dover. Chapter 11: Inner geometry of surfaces. 0-486-63433-7.