Projectively flat surfaces in $\textbf {A}^ 3$

Abstract:

We consider a nondegenerate immersion $f:{M^2} \to {\mathbb {A}^3}$ of an orientable $2$-dimensional manifold ${M^2}$ together with the Blaschke connection $\nabla$ induced on ${M^2}$; this work is aimed at studying locally convex surfaces whose Blaschke connection is projectively flat, reducing the problem of their classification to a system of PDE’s. In particular we can prove the existence of locally convex projectively flat surfaces which are not locally symmetric.