Surfaces containing two isotropic circles through each point

Abstract

We prove (under some technical assumptions) that each surface in $\mathbb R^3$ containing two arcs of parabolas with axes parallel to $Oz$ through each point has a parametrization $\left(\frac{P(u,v)}{R(u,v)},\frac{Q(u,v)}{R(u,v)},\frac{Z(u,v)}{R^2(u,v)}\right)$ for some $P,Q,R,Z\in\mathbb R[u,v]$ such that $P,Q,R$ have degree at most 1 in $u$ and $v$, and $Z$ has degree at most 2 in $u$ and $v$. The proof is based on the observation that one can consider a parabola with vertical axis as an isotropic circle; this allows us to use methods of the recent work by M. Skopenkov and R. Krasauskas in which all surfaces containing two Euclidean circles through each point are classified. Such approach also allows us to find a similar parametrization for surfaces in $\mathbb R^3$ containing two arbitrary isotropic circles through each point (under the same technical assumptions). Finally, we get some results concerning the top view (the projection along the $Oz$ axis) of the surfaces in question.