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}.