Ruled and quadric surfaces in the 3-dimensional Euclidean space satisfying $\Delta ^{III}\boldsymbol{x} = \varLambda \boldsymbol{x}$

TL;DR

This paper classifies ruled and quadric surfaces in 3D Euclidean space that satisfy a specific differential relation involving the third fundamental form, showing that only helicoids and spheres meet this criterion.

Contribution

It proves that the only ruled and quadric surfaces satisfying the relation  extsuperscript{III}oldsymbol{x}=oldsymbol{ extLambda}oldsymbol{x} are helicoids and spheres, providing a complete classification.

Findings

Helicoids satisfy the relation  extsuperscript{III}oldsymbol{x}=oldsymbol{ extLambda}oldsymbol{x.

Spheres satisfy the relation  extsuperscript{III}oldsymbol{x}=oldsymbol{ extLambda}oldsymbol{x.

No other ruled or quadric surfaces satisfy this relation.

Abstract

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation $\Delta ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda $ is a square matrix of order 3. We show that helicoids and spheres are the only surfaces in $E^3$ satisfying the preceding relation.