Affine locally symmetric surfaces (Q1205438)

The author investigates the class of unimodular affine immersions from a surface \(M\) into \(R^ 3\) which induce a locally symmetric Berwald- Blaschke structure on \(M\), with affine shape operator of nonzero rank. Such an immersion \(f: M \to R^ 3\) is firstly characterized by the equivalent conditions that the images of the unimodular affine normal and the conormal vector fields are contained in central quadrics. For the subclass of those affine locally symmetric surfaces whose affine shape operators are diagonalizable, natural isothermal coordinates on \(M\) are introduced in which the affine shape operator is expressed in its Jordan form, the immersion \(f\) is described in those local coordinates and the eigenvalue of the affine shape operator is shown to satisfy some partial differential equations. With a similar approach it is also considered the case in which the affine shape operator is nonsingular and not diagonalizable.