Hokkaido Mathematical Journal

HASEGAWA Kazuyuki,

Affine differential geometry of the unit normal vector fields of hypersurfaces in the real space forms.

Hokkaido Mathematical Journal, 35 (2006) pp.613-627

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Abstract

In this paper, for a hypersurface in the real space form of constant curvature, we prove that the unit normal vector field is an affine imbedding into a certain sphere bundle with canonical metric. Moreover, we study the relations between a hypersurface and its unit normal vector field as an affine imbedding. In particular, several hypersurfaces are characterized by affine geometric conditions which are independent of the choice of the transversal bundle.

MSC(Primary)53C43
MSC(Secondary)53C42
Uncontrolled Keywordssection of sphere bundle, canonical metric, metrically minimal affine immersion, metrically totally umbilic affine immersion