Centroaffine surfaces in ℝ⁴ with planar ∇-geodesics

Scharlach, Christine; Vrancken, Luc

doi:10.1090/S0002-9939-98-04408-6

Centroaffine surfaces in $\mathbb {R}^{4}$ with planar $\nabla$-geodesics
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by Christine Scharlach and Luc Vrancken

Proc. Amer. Math. Soc. 126 (1998), 213-219

DOI: https://doi.org/10.1090/S0002-9939-98-04408-6

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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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