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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Centroaffine surfaces in $\mathbb{R}^{4}$
with planar $\nabla $-geodesics


Authors: Christine Scharlach and Luc Vrancken
Journal: Proc. Amer. Math. Soc. 126 (1998), 213-219
MSC (1991): Primary 53A15; Secondary 53B05, 53B25
MathSciNet review: 1452827
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Abstract: For (positive) definite surfaces in $\mathbb{R}^{4}$ there is a canonical choice of a centroaffine normal plane bundle, which induces a centroaffine invariant Ricci-symmetric connection $\nabla $. We classify all surfaces in $\mathbb{R}^{4}$ with planar $\nabla $-geodesics. It turns out that the resulting class of surfaces is umbilical with projectively flat induced connection and flat normal plane bundle.


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Additional Information

Christine Scharlach
Affiliation: Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: cs@math.tu-berlin.de

Luc Vrancken
Affiliation: Departemente Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium
Email: luc.vrancken@wis.kuleuven.ac.be

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04408-6
PII: S 0002-9939(98)04408-6
Keywords: Centroaffine geometry, centroaffine surfaces in $\mathbb{R}^{4}$, centroaffine normal plane bundle, induced connection, planar geodesics
Received by editor(s): March 19, 1996
Additional Notes: The authors were supported in part by the DFG-project “Affine differential geometry" at the TU Berlin.
The first author was supported in part by the DFG-Forschungsstipendium Scha 698/1-1.
The last author is a Senior Research Assistant of the National Fund for Scientific Research (Belgium).
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society