A Weighted Steiner Minimal Tree for Convex Quadrilaterals on the Two-Dimensional K-Plane

We provide a method to find a weighted Steiner minimal tree for convex quadrilaterals on a two-dimensional hemisphere of radius $\frac{1}{\sqrt{K}}$, for $K>0$ and the two dimensional hyperbolic plane of constant Gaussian Curvature K, for $K<0$ by introducing a method of cyclical differentiation of the objective function with respect to four variable angles. By applying this method, we find a generalized solution to a problem posed by C.F. Gauss in the spirit of weighted Steiner trees.