Consider the helicoid S given by the parametrization x(u,v)=(vcosu, vsinu,u). Find the asymptotic curves on S.
The curvature should be a numeric quantity (scalar) that is positive for a positively oriented space and negative for a negatively oriented space.
Now if you are trying to find the curvature at a typical point where that equals zero then you will need the Bi-Normal, Normal, and Tangent vectors do this calculation.
So you have the normalized length of the normal when you look at what the normalized vector is divided by (SQRT(1 + v^2))
The only way I can see this happening is basically in line with your answer that they are both constant.
If you have something with a zero curvature then it is a plane.
In this case it is a plane parametrized by two parameters u and v and a plane is flat if the tangent vectors for the rate of change of how the plane changes in that direction is zero.
Both tangential vectors (the primary tangent and the secondary tangent or bi-normal) will be constant for a flat space and this is only satisfied if your above conditional holds (i.e. u and v are constant).
If they are allowed to vary then SQRT(1 + v^2) >= 1 for all v so it will never approach zero for all real values of v. If you are considering complex values then this is even more complex (this is a branch known as Kahler geometry with an umlaut a) with completely different conditions.
Now this is for the curvature at a point: if you need to consider other forms of curvature then you will need to consider what those are, but for flat objects the intrinsic curvature as far as I recall should be 0 as well.