Flat affine surfaces in \(\mathbb{R}^4\) with flat normal connection
Due to Burstin, Meyer, Nomizu and Vrancken there exists an affine invariant metric on every nondegenerate surface \(M\) in \(\mathbb{R}^4\). Choosing a special transversal plane bundle an equiaffine structure is defined on \(M\). Discussing surfaces with flat induced connection and trivial normal connection a remarkable difference to the codimension one case turns out: There are such surfaces which do not have constant equiaffine normal bundle. The main theorem gives a complete classification of these surfaces.
scientific article; zbMATH DE number 1505828