AN EXISTENCE THEOREM FOR A SURFACE IN ＄S^{n}＄ WITH A GIVEN MAP AS ITS GAUSS MAP
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概要
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"The purpose of this paper is to study the existence of a surface in the n- dimensional Euclidean unit sphere ＄S^{n}＄ with prescribed Gauss map. For a given ＄0￥infty＄-mapping ＄G＄ from a torus ＄T^{2}＄ into the complex quadric ＄Q_{n-1}＄ , we show that there exists a conformal immersion ＄X＄ : ＄￥hat{T}^{2}￥rightarrow S^{n}＄ such that the Gauss map of the surface ＄S=(T^{2}, S^{n}, X)＄ is ＄ Go￥pi＄ where ＄￥pi＄ : ＄￥hat{T}^{2}￥rightarrow T^{2}＄ is a covering map. Let ＄G＄ be a ＄ c￥infty＄-mapping from a ＄nnected＄ Riemann surface ＄M＄ into ＄Q_{n-1}＄ . Under a certain condition for ＄G＄ we also show that there exists a surface defined by a ＄ o￥infty＄-conformal immersion ＄X＄ from ＄M＄ to the n-dimensional real projective space ＄RP^{n}＄ with the property that a neighborhood of each point of ＄X(M)＄ is covered by a surface in ＄S^{n}＄ with prescribed Gauss map ＄G＄ . By using this result we give a characterization of certain tori immersed in ＄RP^{n}＄ ."
- Yokohama City University and Yokohama National Universityの論文
- 2006-00-00
Yokohama City University and Yokohama National University | 論文
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