Glasnik Matematicki, Vol. 46, No. 2 (2011), 455-469.

TRANSLATION SURFACES IN THE GALILEAN SPACE

Željka Milin Šipuš and Blaženka Divjak

Department of Mathematics, University of Zagreb, Bijenička cesta 30, 10 000 Zagreb, Croatia
e-mail: zeljka.milin-sipus@math.hr

Faculty of organization and informatics, University of Zagreb, Pavlinska 2, 42 000 Varaždin, Croatia
e-mail: blazenka.divjak@foi.hr


Abstract.   In this paper we describe, up to a congruence, translation surfaces in the Galilean space having constant Gaussian and mean curvatures as well as translation Weingarten surfaces. It turns out that, contrary to the Euclidean case, there exist translation surfaces with constant Gaussian curvature K that are not cylindrical surfaces, and translation surfaces with constant mean curvature H ≠ 0 that are not ruled.

2000 Mathematics Subject Classification.   53A35, 53A40.

Key words and phrases.   Galilean space, translation surface, Weingarten surface.


Full text (PDF) (access from subscribing institutions only)

DOI: 10.3336/gm.46.2.14


References:

  1. F. Dillen, W. Goemans and I. Van de Woestyne, Translation surfaces of Weingarten type in 3-space, Bull. Transilv. Univ. Brašov Ser. III 1(50) (2008), 109-122.
    MathSciNet    

  2. L. P. Eisenhart, A treatise on the differential geometry of curves and surfaces, Ginn and company (Cornell University Library), 1909.
    MathSciNet    

  3. M. H. Kim and D. W. Yoon, Weingarten quadric surfaces in a Euclidean 3-space, Turkish J. Math. 35 (2011), 479-485.
    CrossRef

  4. E. Kreyszig, Differential geometry, Dover Publ.Inc., New York, 1991.
    MathSciNet    

  5. H. Liu, Translation surfaces with constant mean curvature in 3-dimensional spaces, J. Geom. 64 (1999), 141-149.
    MathSciNet     CrossRef

  6. R. López, Minimal translation surfaces in hyperbolic space, to appear in Beitrage zur Algebra und Geometrie (Contributions to Algebra and Geometry).

  7. M. Magid and L. Vrancken, Affine translation surfaces with constant sectional curvature, J. Geom. 68 (2000), 192-199.
    MathSciNet     CrossRef

  8. M. I. Munteanu and A. I. Nistor, Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space, in: Proceedings of the VIII International Colloquuim on Differential Geometry, World Scientific, Hackensack, 2009, 316-320.
    MathSciNet     CrossRef

  9. Ž. Milin Šipuš, Ruled Weingarten surfaces in the Galilean space, Period. Math. Hungar. 56 (2008), 213-225.
    MathSciNet     CrossRef

  10. O. Röschel, Die Geometrie des Galileischen Raumes, Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 1985.
    MathSciNet    

  11. D. W. Yoon, Polynomial translation surfaces of Weingarten types in Euclidean 3-space, Cent. Eur. J. Math. 8 (2010), 430-436.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page