Let  $S\subset \mathbb{R}^3$  be  a  smooth surface  with the Gauss normal map $N:S\to  S^2$.

Then for  every $x\in S$, the  differential $(dN)_x:T_xS\to T_{N(x)}S^2$ can be  considered as  an  endomorphism of the  tangent space $T_xS$ since $T_xS$ is  parallel to  $T_{N(x)}S^2$. So  from  now  on, without any  ambiguity and in a  unique way,   we  count $dN$ as an  endomorphism of  the  tangent  bundle $TS$ of  the  surface $S$. So  $dN$  defines  a  linear  operator  $$dN:\chi^{\infty}(S)\to \chi^{\infty}(S)$$  where $\chi^{\infty}(S)$ is the  space  of  all smooth vector  fields on $S$. 

>Under which geometric conditions on $S$ this operator preserve the  Lie  bracket  of  $\chi^{\infty}(S)$? Under which  conditions the  range $dN(\chi^{\infty}(S))$ of this operator is  a  Lie  algebra?