## Osaka Journal of Mathematics

### Codazzi fields on surfaces immersed in Euclidean 4-space

#### Abstract

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

#### Article information

Source
Osaka J. Math., Volume 45, Number 4 (2008), 877-894.

Dates
First available in Project Euclid: 26 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1227708823

Mathematical Reviews number (MathSciNet)
MR2493960

Zentralblatt MATH identifier
1173.53005

#### Citation

Gutiérrez Núñez, J.M.; Romero Fuster, M.C.; Sánchez-Bringas, F. Codazzi fields on surfaces immersed in Euclidean 4-space. Osaka J. Math. 45 (2008), no. 4, 877--894. https://projecteuclid.org/euclid.ojm/1227708823

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