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

DOI:
https://doi.org/10.1090/S0002-9947-1986-0816303-X

MathSciNet review:
816303

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Abstract | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9947-1986-0816303-X

Keywords:
Riemann surface,
Gauss map of a conformal immersion,
Grassmannian,
normal bundle,
tension

Article copyright:
© Copyright 1986
American Mathematical Society