Osaka Journal of Mathematics

Codazzi fields on surfaces immersed in Euclidean 4-space

J.M. Gutiérrez Núñez, M.C. Romero Fuster, and F. Sánchez-Bringas

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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

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

First available in Project Euclid: 26 November 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A05: Surfaces in Euclidean space 57R25: Vector fields, frame fields


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.

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