Does there exist a type of surface such that it has a constant (non-zero) gradient of gaussian curvature? If a class of surfaces (2D Riemannian Manifolds) with constant gradient of Gaussian curvature exists, what is it called and how is it classified? Are there maybe only surfaces that have this property in a certain region?
Edit:
By constant gradient, I guess the way to formalise it would be to say that $\partial_a\partial_b K = 0$. Would that in some sense mean its gradient is constant?
 A: Your notation $\partial_a \partial_b K=0$ is unclear to me. I think, what you mean is that the vector field $Y=\operatorname{grad} K$ is parallel, i.e. $\nabla_X Y=0$ for every vector field $X$ on your surface. Then such metrics (with nonzero $Y$) do not exist. Indeed, if $(M,g)$ is an $m$-dimensional Riemannian manifold which admits a nonzero parallel vector field $Y$, then $(M,g)$ locally splits as the product of the real line and  a Riemannian manifold of dimension $m-1$ (the real line factor corresponds to the flow-lines of $Y$), see here. In the case when $M$ is a surface, this means that $(M,g)$ is locally a product of two 1-dimensional Riemannian manifolds, i.e. is locally flat. This, of course, implies that $Y=\operatorname{grad} K=0$, so our  parallel vector field was zero to begin with.
A small modification of this argument works in higher dimensions and one obtains:
Suppose that $(M,g)$ is a Riemannian manifold with parallel gradient field of the scalar curvature function $R_g$. Then $R_g$ is constant.
