Document Type : Research Paper


1 Faculty of Basic Sciences, University of Bonab, , P.O.Box 5551761167, Bonab, Iran.

2 Department of Mathematics, Faculty of Science, University of ABCD, P.O.Box xxxx, City, Country.


Commutative properties of four-dimensional generalized symmetric pseudo-Riemannian manifolds were considered. Specially, in this paper, we studied Skew-Tsankov and Jacobi-Tsankov conditions in 4-dimensional pseudo-Riemannian generalized symmetric manifolds.


[1] M. Brozos-Vazquez, E. Garcia, P. Gilkey and R. -Vazquez-Lorenzo, Examples of signature (2, 2)-manifolds with commuting curvature operators, J. Phys. A: Math. Theor. 40 (2007) 13149{13159.
[2] M. Brozos-Vazquez and P. Gilkey, The global geometry of Riemannian manifolds with commuting curvature operators, J. Fixed Point Theory Appl. 1 (2007) 87-96.
[3] M. Brozos-Vazquez and P. Gilkey, Manifolds with commuting Jacobi operators, J. Geom. 86 (2007) 21-30.
[4] G. Calvaruso, Harmonicity of vector elds on four-dimensional generalized symmetric spaces, Cent. Eur. J. Math. 10 (2012), 411-425.
[5] G. Calvaruso and B. De Leo, Curvature Properties of Four-Dimensional Generalized Symmetric Spaces, J. Geom. 90 (no. 1-2) (2008), 30-46.
[6] G. Calvaruso and A. H. Zaeim, Geometric Structures over Four-Dimensional Generalized Symmetric Spaces, Mediterr. J. Math. 10 (2013), 971-987.
[7] J.  Cerny and O. Kowalski, Classi cation of generalized symmetric pseudo-Riemannian spaces of dimension n  4, Tensor (N.S.) 38 (1982), 256-267.
[8] E. Garcia-Rio, A. Haji-Badali, M. E. Vazquez-Abal and R. Vazqes-Lorenzo, Lorentzian 3-manifold with commuting curvature operators, Int. J. Geom. Meth. Modern Phys. 5 (4) (2008), 557-572.
[9] P. Gilkey, Geometric Properties of neutral Operators De ned by the Riemannian Curvature Tensor World Scienti c Publishing Co., Inc., River Edge, NJ, 2001.
[10] C. Gonzalez and D. Chinea, Estructuras homogeneas sobre espacios simetricos generalizados, Proceedings of the XIIth Portuguese-Spanish Conference on Mathematics, Vol. II, 572{578, Univ. Minho, Braga, 1987.
[11] B. Komrakov Jnr., Einstein-Maxwell equation on four-dimensional homogeneous spaces, Lobachevskii J. Math. 8 (2001), 33-165.
[12] D. Kotschick and S. Terzic, On formality of generalized symmetric spaces, Math. Proc. Cambridge Philos. Soc. 134 (2003), 491-505.
[13] O. Kowalski, Generalized symmetric spaces, Lectures Notes in Math. 805, Springer-Verlag, Berlin-New York, 1980.
[14] B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983.
[15] S. Terzic, Real cohomology of generalized symmetric spaces, Fundam. Prikl. Mat. 7(2001), 131-157.
[16] S. Terzic, Pontryagin classes of generalized symmetric spaces (Russian), Mat. Zametki 69 (2001), 613{621. Translation in Math. Notes 69 (no. 3{4) (2001), 559-566.
[17] Y. Tsankov, A characterization of n-dimensional hypersurface in Euclidean space with commuting curvature operators, Banach Center Publ. 69 (2005) 205-209.