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

Author(s): Christine Scharlach; Luc Vrancken
Journal: Proc. Amer. Math. Soc. 126 (1998), 213-219.
MSC (1991): Primary 53A15; Secondary 53B05, 53B25
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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.

References:

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K. Nomizu and T. Sasaki, Centroaffine immersions of codimension two and projective hypersurface theory, Nagoya Math. J. 132 (1993), 63-90. MR 94j:53013

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K. Nomizu and L. Vrancken, A new equiaffine theory for surfaces in $\mathbb{R}^{4}$, Internat. J. Math. 4 (1993), 127-165. MR 94f:53014

[Sch1]
Ch. Scharlach, Centroaffine differential geometry of surfaces in $\mathbb{R}^{4}$, Dissertation, Technische Universität Berlin, 1994.

[Sch2]
Ch. Scharlach, Centroaffine first order invariants of surfaces in $\mathbb{R}^{4}$, Results Math. 27 (1995), 141-159. MR 96a:53016

[Sch3]
Ch. Scharlach, Centroaffine differential geometry of (positive) definite oriented surfaces in $\mathbb{R}^{4}$, Preprint No. 451/1995, FB Mathematik, TU Berlin, submitted.

[Vr]
L. Vrancken, Affine surfaces whose geodesics are planar curves, Proc. Amer. Math. Soc. 123 (1995), 3851-3854. MR 96b:53021

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Additional Information:

Christine Scharlach
Affiliation: Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Stra{ß}e 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: 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
Copyright of article: Copyright 1998, American Mathematical Society

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