A Classification of Strict Walker 3-Manifold

Year 2021, Volume: 9 Issue: 1, 148 - 153, 28.04.2021

Athoumane Nıang Ameth Ndiaye Abdoul Salam Diallo

Abstract

In this paper we give two special families of ruled surfaces in a three dimensional strict Walker manifold. The local degeneracy (resp. non-degeneracy) of one of this family has a strong consequence on the geometry of the ambiant Walker manifold.

Keywords

Minimal surfaces, Lorentz Manifold, Ruled surfaces

References

• [1] M. Brozos-Vàzquez, E. Garca-Rio, P. Gilkey, S. Nikevic and R. Vazquez-Lorenzo. The Geometry of Walker Manifolds, Synthesis Lectures on Mathematics and Statistics, 5. Morgan and Claypool Publishers, Williston, VT, 2009.
• [2] M. P. do Carmo, Differential geometry of curves and surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976. viii+503 pp. 190-191.
• [3] M. Chaichi, E. Garcia-Rio and M.E. Vàzquez-Abal, Three-dimensional Lorentz manifolds admitting a parallel null vector eld, J. Phys. A: Math. Gen. 38 (2005), 841-50.
• [4] A. S. Diallo and F. Massamba, Some properties of four-dimensional Walker manifolds, New Trends Math. Sci, 5 (2017), (3), 253-261.
• [5] A. S. Diallo, A. Ndiaye and A. Niang, Minimal graphs on three-dimensional Walker manifolds, To appear.
• [6] G. Calvaruso and J. Van der Veken, Parallel surfaces in Lorentzian three-manifolds admitting a parallel null vector field, J. Phys. A: Math. Theor. 43 (2010) 325-207.
• [7] A. Niang, Surfaces minimales réglées dans l'espace de Minkowski ou Euclidien orienté de dimension 3, Afrika Mat. 15 (2003), (3), 117-127.
• [8] K. Nomizu and T. Sasaki, Affne Differential Geometry. Geometry of Affne Immersions. Cambridge Tracts in Mathematics Vol. 111 (Cambridge University Press, Cambridge, (1994).
• [9] B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, (1983).
• [10] A. G. Walker, Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math. Oxford 1 (1950), (2), 69-79.
• [11] I. Van de Woerstyne, Minimal surfaces of the 3-dimensional Minkowski space, Geometry and topology of submanifolds, II (Avignon, 1988), 344-369.