Affine surfaces which admit several affine immersions in $\mathbb{R}^3$

Abstract

Let $F: \Sigma \to \mathbb{R}^3$ be a Blaschke immersion of an affine surface $(\Sigma,\nabla)$ with a positive definite affine fundamental form such that $dim Im \, R = 1$ where $R$ is the curvature tensor. Suppose that there exists another immersion of the same surface with the same induced affine connection $\nabla$ which is not affine equivalent to the first one. Then we give explicitely $F$. Therefore all immersions which admit another immersion which is not affine equivalent to the original one are classified.