Parallel affine immersions \(M^n\to\mathbb R^{n+2}\) with flat connection
An affine immersion is called \textit{parallel}, if the cubic form vanishes identically. The paper under review is devoted to the study of parallel affine immersions \(f:M^n\rightarrow \mathbb R^{n+2}\) with flat connection. The so called \textit{point codimension \(q\)} is defined as the rank of the affine fundamental form and does not depend on the choice of the transverse distribution. The cases possible are \(q=0,1,2\). While the first one gives an affine subspace of \(\mathbb R^n\), the second one leads to the hypersurface situation and thus to hyperquadrics. So the main discussion is the solution in case \(q=2\), where a complete classification is given, depending on the rank of the Weingarten mapping.
scientific article; zbMATH DE number 6508314