On Finite-Type Gauss Maps of Quadric Surfaces in Euclidean 3-Space

This paper focuses on the study of quadric surfaces in three-dimensional Euclidean space that satisfy the finite II-type Gauss map condition, a concept introduced firstly by B.-Y. Chen in its first fundamental form; later it was applied by other researchers in the second and third fundamental forms. This study’s primary finding is that spheres are the only quadric surfaces with this characteristic. This suggests a particular and significant grouping in the more general class of quadric surfaces according to their finite-type properties with respect to the second fundamental form.