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3-dimensional affine hypersurfaces in ℝ4 with parallel cubic form

Published online by Cambridge University Press:  22 January 2016

Franki Dillen  and
Luc Vrancken
Franki Dillen
Affiliation:
Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B B-3001 Leuven Belgium
Luc Vrancken
Affiliation:
Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B B-3001 Leuven Belgium
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In this paper, we study 3-dimensional locally strongly convex affine hypersurfaces in ℝ4. Since the publication of Blaschke’s book [B] in the early twenties, it is well-known that on a nondegenerate affine hyper-surface M there exists a canonical transversal vector field called the affine normal. The second fundamental form associated to the affine normal is called the affine metric. In the special case that M is locally strongly convex, this affine metric is a Riemannian metric. Also, using the affine normal, by the Gauss formula one can introduce an affine connection on M, called the induced connection ∇. So on M, we can consider two connections, namely the induced affine connection ∇ and the Levi Civita connection of the affine metric h.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 124 , December 1991 , pp. 41 - 53
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

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