Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Symmetry Properties of 3-dimensional D'Atri Spaces

  • Belkhelfa, Mohamed (Laboratoire de Physique Quantique de la Matiere et Modelisations, Mathematiques - L.P.Q. 3M, Centre Universitaire) ;
  • Deszcz, Ryszard (Department of Mathematics, Agricultural University of Wroclaw) ;
  • Verstraelen, Leopold (Department of Mathematics, Catholic University Leuven)
  • Received : 2005.02.17
  • Published : 2006.09.23
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Abstract

We investigate semi-symmetry and pseudo-symmetry of some 3-dimensional Riemannian manifolds: the D'Atri spaces, the Thurston geometries as well as the ${\eta}$-Einstein manifolds. We prove that all these manifolds are pseudo-symmetric and that many of them are not semi-symmetric.

Keywords

References

  1. C. Baikoussis and D. E. Blair, On Legendre curves in contact 3-dimensional manifolds, Geometria Dedicata, 49(1994), 135-142. https://doi.org/10.1007/BF01610616
  2. M. Belkhelfa, R. Deszcz, M. Glogowska, M. Hotlos, D. Kowalczyk, and L. Verstraelen, On some type of curvature conditions, in: Banach Center Publ., Inst. Math. Polish Acad. Sci., 57(2002), 179-194.
  3. L. Bianchi, Lezioni sulla teoria dei gruppi continui di transformazioni, Editioni Zanichelli, Bologna, 1928.
  4. D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Math., vol. 509, Springer-Verlag, 1976.
  5. D. E. Blair, T. Koufogiorgos, and S. Ramesh, A classification of 3-dimensional contact metric manifolds with $Q{\phi}={\phi}Q$, Kodai Math. J., 13(3)(1990), 391-401. https://doi.org/10.2996/kmj/1138039284
  6. F. Borghero and R. Caddeo, Une structure de separabilite et geodesiques dans les huit geometries tridimensionelles de Thurston, Rend. Mat. Appl Ser., 9(4)(1989), 607-624.
  7. R. Caddeo, P. Piu, and A. Ratto, So(2)-invariant minimal and constant mean curvature surfaces in 3-dimensional homogeneous spaces, Manuscripta Math., 87(1995), 1-12. https://doi.org/10.1007/BF02570457
  8. E. Cartan, Lecons sur la geometrie des espaces de Riemann, second ed., Gauthier-Villards Paris, 1946.
  9. J. E. D'Atri and H. K. Nickerson, Divergence preserving geodesic symmetries, J. Diff. Geom., 3(1969), 467-476. https://doi.org/10.4310/jdg/1214429067
  10. R. Deszcz, On pseudosymmetric spaces, Bull. Soc. Math. Belg., 44(ser. A)(1992), 1-34.
  11. R. Deszcz, Curvature proprerties of certain compact pseudosymmetric manifolds, Colloq. Math., 65(1993), 139-147. https://doi.org/10.4064/cm-65-1-139-147
  12. R. Deszcz, On pseudosymmetric manifolds, Dept. Math., Agricultural Univ.Wroclaw, Ser. A, Theory and Methods, Report No. 34, (1995).
  13. R. Deszcz, On the equivalence of Ricci-semisymmetry and semisymmetry, Dept. Math., Agricultural Univ. Wroclaw, Ser. A, Theory and Methods, Report, No. 64, (1998).
  14. R. Deszcz, L. Verstraelen, and S. Yaprak, Pseudosymmetric hypersurfaces in 4- dimensional spaces of constant curvature, Bull. Inst. Math. Acad. Sinica, 22(1994), 167-179.
  15. J. Inoguchi, Minimal surfaces in 3-dimensional solvable Lie groups, Chin. Ann. Math., 24B(2003), 73-84.
  16. O. Kowalski, Space with volume preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Math. Torino, Fasc. Spec., (1983), 131-159.
  17. S. Maier, Conformal flatness and self-duality of Thurston-geometries, Proc. Am. Math. Soc., 4(1998), 1165-1172.
  18. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  19. D. Perrone, Torsion and critical metrics on contact three-manifolds, Kodai Math. J., 13(1990), 88-100. https://doi.org/10.2996/kmj/1138039163
  20. P. Piu, Sur certain types de distributions non integrables totalement geodesiques, Ph.D. thesis.
  21. P. Scott, The geometries of 3-manifolds, Bull. Lond. Math. Soc., 15(1983), 401-487. https://doi.org/10.1112/blms/15.5.401
  22. W. M. Thurston, Three-dimensional Geometry and Topology, Silvio Levy ed., vol. 1, Princeton University Press, 1997.
  23. L. Verstraelen, Comments on pseudo-symmetry in sense of R. Deszcz, in: Geometry and Topology of Submanifolds, World Sci. Singapore, VI(1994), 199-209.
  24. G. Vranceanu, Lecons de geometrie differentielle, Acad. R. P. Roumanie ed., 1957.
  25. T. J. Willmore, Riemannian Geometry, Clarendon Press, Oxford, 1993.
  26. S. Yaprak, Intrinsic and extrinsic differential geometry concerning conditions of pseudo-symmetry, Ph.D. thesis, K.U.Leuven, 1993.