Tokyo Journal of Mathematics

Surfaces in $S^{n}$ with Prescribed Gauss Map

Ayako Tanaka

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Let $G$ be a $C^{\infty }$-mapping from a connected Riemann surface $M$ into the complex quadric $Q_{n-1}$ in the $n$-dimensional complex projective space. We give a condition for the existence of a surface in the $n$-dimensional Euclidean unit sphere $S^{n}$ such that the Gauss map is $G$. Under this condition, if $M$ is a torus, there exists a surface in $S^{n}$ such that the Gauss map is $G$. We also show that for a connected Riemann surface $M$ there exists an immersion $X:M\rightarrow RP^{n}$ such that a neighborhood of each point of $X(M)$ is covered by a surface in $S^{n}$ with prescribed Gauss map $G$ where $RP^{n}$ is the $n$-dimensional real projective space.

Article information

Tokyo J. Math., Volume 29, Number 1 (2006), 91-110.

First available in Project Euclid: 20 December 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]


Tanaka, Ayako. Surfaces in $S^{n}$ with Prescribed Gauss Map. Tokyo J. Math. 29 (2006), no. 1, 91--110. doi:10.3836/tjm/1166661869.

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