The Gauss map for surfaces. II. The Euclidean case
Author:
Joel L. Weiner
Journal:
Trans. Amer. Math. Soc. 293 (1986), 447466
MSC:
Primary 53A07; Secondary 53A05
MathSciNet review:
816303
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Abstract: We study smooth maps of a Riemann surface into the Grassmannian of oriented planes in and determine necessary and sufficient conditons on in order that it be the Gauss map of a conformal immersion . We sometimes view as an oriented riemannian vector bundle; it is a subbundle of , the trivial bundle over with fibre . The necessary and sufficient conditions obtained for simply connected involve the curvatures of and , the orthogonal complement of in , as well as certain components of the tension of viewed as a map , where is a unit sphere of dimension that contains as a submanifold in a natural fashion. If satisfies a particular necessary condition, then the results take two different forms depending on whether or not is the Gauss map of a conformal minimal immersion. The case is also studied in some additional detail.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719860816303X
PII:
S 00029947(1986)0816303X
Keywords:
Riemann surface,
Gauss map of a conformal immersion,
Grassmannian,
normal bundle,
tension
Article copyright:
© Copyright 1986
American Mathematical Society
