On rigidity of affine surfaces

Rigidity of nondegenerate Blaschke surfaces in $\mathbf {R}^{3}$ is studied. The rigidity criteria are given in terms of $\nabla R$, where $R$ is the curvature of the Blaschke connection $\nabla$. If the rank of $\nabla R$ is 2, then the surface is rigid. If $\nabla R=0$, it is nonrigid. In the case where the rank of $\nabla R$ is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.