Consider a Riemannian vector bundle of rank 1 defined by a
normal vector field $\nu$ on a surface $M$ in $\mathbb{R}^{4}$.
Let $\mathrm{II}_{\nu}$ be the second fundamental form with
respect to $\nu$ which determines a configuration of lines
of curvature. In this article, we obtain conditions on $\nu$
to isometrically immerse the surface $M$ with $\mathrm{II}_{\nu}$
as a second fundamental form into $\mathbb{R}^{3}$. Geometric
restrictions on $M$ are determined by these conditions. As
a consequence, we analyze the extension of Loewner's conjecture,
on the index of umbilic points of surfaces in $\mathbb{R}^{3}$,
to special configurations on surfaces in $\mathbb{R}^{4}$.