Codazzi fields on surfaces immersed in Euclidean 4-space

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 $\mathbb{R}^{4}$.