Two families of affine projectively flat surfaces
In affine differential geometry, it follows from the basic existence and uniqueness theorem that the conjugate connection is projectively flat. Therefore, by assuming that also the induced connection is projectively flat, one obtains dual immersions with the same affine metric (but with the induced connection and conormal connection interchanged). In the present paper, the author studies affine surfaces whose induced connection is projectively flat and for which the shape operator is nondiagonalisable. Besides ruled surfaces with constant mean curvature, he obtains 3 special classes of affine surfaces. Surprisingly, all of the examples in these families are homogeneous with respect to the full affine group of transformations. The case of projectively flat surfaces with diagonalisable shape operator is considered in [\textit{F. Podesta}, Proc. Am. Math. Soc. 119, 255-260 (1993; Zbl 0790.53010)].
scientific article; zbMATH DE number 1000350