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## Affine surfaces in 4-dimensional affine space with planar geodesics

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Author(s) : Luc Vrancken

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 663-2000

MSC 2000

53A15 Affine differential geometry

Abstract :
In this paper we study nondegenerate affine immersions, as introduced by Nomizu and Pinkall, of an affine surface $(M,\nabla)$ in the 4-dimensional affine space $(\mathbb{R}^4,D)$. Using an existence and uniqueness theorem, we classify all such immersions $\phi$ which map the $\nabla$-geodesics of $M$ into planar curves in $\mathbb{R}^4$. These theorems generalize results previously obtained in the Riemannian, the equiaffine and the centroaffine case.