  A rather natural construction for a smooth random surface in space is the
level surface of value zero, or 'nodal' surface f(x,y,z)=0, of a (real) random
function f; the interface between positive and negative regions of the
function. A physically significant local attribute at a point of a curved
surface is its Gaussian curvature (the product of its principal curvatures)
because, when integrated over the surface it gives the Euler characteristic.
Here the probability distribution for the Gaussian curvature at a random point
on the nodal surface f=0 is calculated for a statistically homogeneous
('stationary') and isotropic zero mean Gaussian random function f. Capitalizing
on the isotropy, a 'fixer' device for axes supplies the probability
distribution directly as a multiple integral. Its evaluation yields an explicit
algebraic function with a simple average. Indeed, this average Gaussian
curvature has long been known. For a non-zero level surface instead of the
nodal one, the probability distribution is not fully tractable, but is supplied
as an integral expression.
