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Hypersurfaces with pointwise 1-type Gauss map. (English) Zbl 1136.53015

Summary: We prove that an oriented hypersurface \(M\) of a Euclidean space \(E^{n+1}\) has pointwise 1-type Gauss map of the first kind if and only if \(M\) has constant mean curvature. Then we conclude that all oriented isoparametric hypersurfaces of \(E^{n+1}\) have 1-type Gauss map. We also show that a rational hypersurface of revolution in a Euclidean space \(E^{n+1}\) has pointwise 1-type Gauss map of the second kind if and only if it is a right \(n\)-cone.

MSC:

53B25 Local submanifolds
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