Gilkey, P.Valle Regueiro, Xabier2020-04-062020-04-062019Gilkey, P. and Valle-Regueiro, X., 2019. Applications of PDEs to the study of affine surface geometry. Matematički Vesnik, 71(1-2), 45-620025-5165http://hdl.handle.net/10347/21178If 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.eng© 2019 by de authors. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) lisense (https://creativecommons.org/licenses/by/4.0/)https://creativecommons.org/licenses/by/4.0/Type A affine surfaceQuasi-Einstein equationAffine Killing vector fieldLocally homogeneous affine surfacejournal article2406-0682open access