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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Gauss map for surfaces. II. The Euclidean case

Author: Joel L. Weiner
Journal: Trans. Amer. Math. Soc. 293 (1986), 447-466
MSC: Primary 53A07; Secondary 53A05
MathSciNet review: 816303
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Abstract: We study smooth maps $ t:\;M \to G_2^c$ of a Riemann surface $ M$ into the Grassmannian $ G_2^c$ of oriented $ 2$-planes in $ {{\mathbf{E}}^{2 + c}}$ and determine necessary and sufficient conditons on $ t$ in order that it be the Gauss map of a conformal immersion $ X:\;M \to {{\mathbf{E}}^{2 + c}}$. We sometimes view $ t$ as an oriented riemannian vector bundle; it is a subbundle of $ {\mathbf{E}}_M^{2 + c}$, the trivial bundle over $ M$ with fibre $ {{\mathbf{E}}^{2 + c}}$. The necessary and sufficient conditions obtained for simply connected $ M$ involve the curvatures of $ t$ and $ {t^ \bot }$, the orthogonal complement of $ t$ in $ {\mathbf{E}}_M^{2 + c}$, as well as certain components of the tension of $ t$ viewed as a map $ t:\;M \to {S^C}(1)$, where $ {S^C}(1)$ is a unit sphere of dimension $ C$ that contains $ G_2^c$ as a submanifold in a natural fashion. If $ t$ satisfies a particular necessary condition, then the results take two different forms depending on whether or not $ t$ is the Gauss map of a conformal minimal immersion. The case $ t:\;M \to G_2^2$ is also studied in some additional detail.

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Keywords: Riemann surface, Gauss map of a conformal immersion, Grassmannian, normal bundle, tension
Article copyright: © Copyright 1986 American Mathematical Society