A Tzitzeica surface has the property that the ratio of the surface’s Gaussian curvature and the fourth power of the distance from the origin to the tangent plane at any arbitrary point of the surface is constant.
My question is: are there Tzitzeica surfaces with constant negative Gaussian curvature?
The answer is 'no'.
Suppose that $M\subset\mathbb{E}^3$ is a smooth connected surface. If the ratio of the Gauss curvature $K$ and $p^4$ is constant (where $p(x)$ is the distance from $T_xM$ to the origin is constant) and $K$ is constant and nonzero, then $p$ is also constant. However, if $p$, which is known as the support function of the surface, is constant and the Gauss curvature $K$ is nonzero, then the surface is a portion of a sphere centered at the origin, and hence $K$ is a positive constant.
