Centroaffine surfaces in ℝ⁴ with planar ∇-geodesics

Abstract

For (positive) definite surfaces in R 4 \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 R 4 \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.