Let M be a surface in $R^3$. $E_1, E_2, E_3$ are orthonormal vector fields defined on M.
I wonder how one can extend say $E_1$ to some open set of $R^3$.
Naturally one wants to move $E_1$ along the normal vector of M at each point. However, those normal vectors at different points may intersect making the new extended vector fields ill defined.
How is such extension always possible?