# Characterization of quadric surfaces in terms of coordinate finite type Gauss map

## TL;DR

This paper characterizes certain quadric surfaces in Euclidean 3-space based on a specific differential relation involving their Gauss map, identifying planes, spheres, and cylinders as unique solutions.

## Contribution

It proves that only planes, spheres, and circular cylinders among quadrics satisfy a particular Laplace-Beltrami relation with their Gauss map.

## Key findings

- Planes, spheres, and cylinders satisfy the relation ^{I} G = M G.
- Other quadric surfaces do not satisfy this relation.
- The relation characterizes these surfaces uniquely among quadrics.

## Abstract

In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space $\mathbb{E}^{3}$. We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map $\boldsymbol{G}$ satisfies a relation of the form $\Delta^{I}\boldsymbol{G}= M \boldsymbol{G}$, where $M$ is a square matrix of order 3 and $\Delta^{I}$ is the Laplace-Beltrami operator corresponding to the first fundamental form $I$ of the surface.