Applications of PDEs to the study of affine surface geometry

Abstract

If M=(M,∇) is an affine surface, let Q(M):=ker(H+1m−1ρs) be the space of solutions to the quasi-Einstein equation for the crucial eigenvalue. Let M~=(M,∇~) be another affine structure on M which is strongly projectively flat. We show that Q(M)=Q(M~) if and only if ∇=∇~ and that Q(M) is linearly equivalent to Q(M~) if and only if M is linearly equivalent to M~. We use these observations to classify the flat Type A connections up to linear equivalence, to classify the Type A connections where the Ricci tensor has rank 1 up to linear equivalence, and to study the moduli spaces of Type A connections where the Ricci tensor is non-degenerate up to affine equivalence.