On rigidity of affine surfaces

Abstract

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