## Tokyo Journal of Mathematics

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

Ayako Tanaka

#### 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

https://projecteuclid.org/euclid.tjm/1166661869

Digital Object Identifier
doi:10.3836/tjm/1166661869

Mathematical Reviews number (MathSciNet)
MR2258274

Zentralblatt MATH identifier
1107.53039

#### 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

#### References

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• F. W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman, Chicago (1971).