Hostname: page-component-84b7d79bbc-lrf7s Total loading time: 0 Render date: 2024-07-26T12:29:52.959Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[B] Blaschke, W., Vorlesungen über Differential geometrie II, Affine Differential-geometrie, Springer, Berlin, 1923.Google Scholar
[BNS] Bokan, N., Nomizu, K. and Simon, U., Affine hypersurfaces with parallel cubic forms, Tôhoku Math. J., 42 (1990), 101–108.Google Scholar
[DV] Dillen, F. and Vrancken, L., Generalized Cayley surfaces, Proceedings of the Conference on Global Analysis and Global Differential Geometry, Berlin 1990, Lecture Notes in Mathematics, Springer Verlag, Berlin.Google Scholar
[KN] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, Volume 1, Interscience Publishers, New York.Google Scholar
[MN] Magid, M. and Nomizu, K., On affine surfaces whose cubic forms are parallel relative to the affine metric, Proc. Nat. Acad. Sci. Ser. A, 65 (1989), 215–218.Google Scholar
[N] Nomizu, K., Introduction to affine differential geometry, part I, MPI/88–37, Bonn (1988).Google Scholar
[NP1] Nomizu, K. and Pinkall, U., On the geometry of affine immersions, Math. Z., 195 (1987), 165–178.Google Scholar
[NP2] Nomizu, K. and Pinkall, U., Cayley surfaces in affine differential geometry, Tôhoku Math. J., 41 (1989), 589–596.Google Scholar
[VI] Vrancken, L., Affine higher order parallel hypersurfaces, Ann. Fac. Sci. Toulouse, 9 (1988), 341–353.CrossRefGoogle Scholar
[V2] Vrancken, L., Affine surfaces with higher order parallel cubic form, Tôhoku Math. J., 43 (1991),127–139.Google Scholar
[VLS] Vrancken, L., Li, A. M. and Simon, U., Affine spheres with constant affine sectional curvature, Math. Z., 206 (1991), 651–658.Google Scholar
[Y] Yu, J. H., Affine hyperspheres with constant sectional curvature in A4, preprint, Sichuan University.Google Scholar