Tokyo Journal of Mathematics

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

Ayako Tanaka

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Abstract

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

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

Dates
First available in Project Euclid: 20 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1166661869

Digital Object Identifier
doi:10.3836/tjm/1166661869

Mathematical Reviews number (MathSciNet)
MR2258274

Zentralblatt MATH identifier
1107.53039

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

Citation

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


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References

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  • D. A. Hoffman and R. Osserman, The Gauss map of surfaces in $R^n$, J. Differential Geometry, 18 (1983), 733–754.
  • F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman, Chicago (1971).