Projectively flat affine surfaces

Abstract

We study non-degenerate affine surfaces in A 3 with a projectively flat induced connection. The curvature of the affine metricK, the affine mean curvature H , and the Pick invariant J are related byK = H + J . Depending on the rank of the span of the gradients of these functions, a local classification of three groups is given. The main result is the characterization of the projectively flat but not locally symmetric surfaces as a solution of a system of ODEs. In the final part, we classify projectively flat and locally symmetric affine translation surfaces. (2000): Primary 53A15; Secondary 53B05. Mathematics Subject Classification

Linear Weingarten centroaffine translation surfaces in R3

“…This class of surfaces has been studied previously by Pabel [15], Binder [1], Magid and Vrancken [14], and other geometers. All classes mentioned are very rich.…”

Affine Translation Surfaces with Constant Gaussian Curvature

“…Later on, Sun and Chen extended this into the case of hypersurfaces [6]. On the other hand, Binder [1] classified locally symmetric affine translation surfaces in R 3 . Here we give a complete classification of nondegenerate affine translation surfaces with constant Gaussian curvature in R 3 .…”

Centroaffine Translation Surfaces in $${\mathbb{R}}^3$$

“…This class of surfaces has been studied previously by Pabel [15], Binder [1], Magid and Vrancken [14], and other geometers. Pabel has discussed translation surfaces with equiaffine methods in affine 3-space R 3 .…”