Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T15:35:21.674Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of affine ruled surfaces

Published online by Cambridge University Press:  18 May 2009

Włodzimierz Jelonek
Włodzimierz Jelonek
Affiliation:
Institute of Mathematics, Technical University of Cracow, Warszawska 24, 31-155 Krakow, Poland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 ∇.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 39 , Issue 1 , January 1997 , pp. 17 - 20
Copyright
Copyright © Glasgow Mathematical Journal Trust 1997

References

REFERENCES

1.Blaschke, W., Vorlesungen über Differentialgeometrie II (Springer, Berlin, 1923).Google Scholar
2.Dillen, F., Locally symmetric complex affine hypersurfaces, J. Geometry 33 (1988), 27–38.Google Scholar
3.Dillen, F., Martinez, A., Milan, F., Santos, F. G. and Vrancken, L., On the Pick invariant, the affine mean curvature and the Gauss curvature of affine surfaces Results in Math. 20 (1991), 622–642.CrossRefGoogle Scholar
4.Jelonek, W., Affine surfaces with parallel shape operators, Ann. Polon. Math. LVI.2. (1992), 179–186.CrossRefGoogle Scholar
5.Jelonek, W.. The local structure of affine spheres in ℝ3, Preprint 505, Polish Academy of Sciences (1992).Google Scholar
6.Jelonek, W., Projectively flat connections and rigidity of affine spheres in Selected problems of Mathematics, Cracow University of Technology, Vol. 6 (1995), 161–169.Google Scholar
7.Jelonek, W., Ruled surfaces in affine differential geometry (to appear in Demonstratio Math.)Google Scholar
8.Magid, M. and Nomizu, K., On affine surfaces whose cubic forms are parallel relative to the affine metric, Proc. Japan Acad. 65, Ser. A (1989), 215–218.Google Scholar
9.Martinez, A and Millan, F., On affine maximal ruled surfaces, Math. Z. 208 (1991), 635–644.CrossRefGoogle Scholar
10.Nomizu, K. and Pinkall, U., On the Geometry of Affine Immersions, Math. Z. 195 (1987), 165–178.CrossRefGoogle Scholar
11.Schirokov, P. A. and Schirokov, A. P., Affine Differentialgeometrie (Teubner, Leipzig, 1962).Google Scholar
12.Vrancken, L., Affine surfaces with constant affine curvatures, Geom. Dedicata 33 (1990), 177–194.CrossRefGoogle Scholar