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Centroaffine surfaces in -geodesics

Authors:	Christine Scharlach Luc Vrancken

Institution:	Fachbereich Mathematik, MA 8-3, Technische Universität Berlin, Stra{ß}e des 17. Juni 136, D-10623 Berlin, Germany ; Departemente Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200 B, B-3001 Leuven, Belgium

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.

Keywords:	Centroaffine geometry centroaffine surfaces in $\mathbb{R}^{4}$ centroaffine normal plane bundle induced connection planar geodesics

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