Geodesic
Characterization of affine ruled surfaces
Glasgow mathematical journal, Tome 39 (1997) no. 1, pp. 17-20

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The aim of this paper is to give certain conditions characterizing ruled affine surfaces in terms of the Blaschke structure (∇, h, S) induced on a surface (M, f) in R3. The investigation of affine ruled surfaces was started by W. Blaschke in the beginning of our century (see [1]). The description of affine ruled surfaces can be also found in the book [11], [3] and [7]. Ruled extremal surfaces are described in [9]. We show in the present paper that a shape operator S is a Codazzi tensor with respect to the Levi-Civita connection ∇ of affine metric h if and only if (M, f) is an affine sphere or a ruled surface. Affine surfaces with ∇S = 0 are described in [2] (see also [4]). We also show that a surface which is not an affine sphere is ruled iff im(S - HI) =ker(S - HI) and ket(S - HI) ⊂ ker dH. Finally we prove that an affine surface with indefinite affine metric is a ruled affine sphere if and only if the difference tensor K is a Codazzi tensor with respect to ∇. 
Jelonek, Włodzimierz. Characterization of affine ruled surfaces. Glasgow mathematical journal, Tome 39 (1997) no. 1, pp. 17-20. doi: 10.1017/S0017089500031852
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