A Geometrical Characterization of Spatially Curved Roberstion-Walker Space and Its Retractions


Our aim in the present article is to introduce and study new types of retractions of closed flat Robertson-Walker W4 model. Types of the deformation retract of closed flat Robertson-Walker W4 model are obtained. The relations between the retraction and the deformation retract of curves in W4 model are deduced. Types of minimal retractions of curves in W4 model are also presented. Also, the isometric and topological folding in each case and the relation between the deformation retracts after and before folding have been obtained. New types of homotopy maps are deduced. New types of conditional folding are presented. Some commutative diagrams are obtained.

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A. El-Bagoury and A. Al-Luhaybi, "A Geometrical Characterization of Spatially Curved Roberstion-Walker Space and Its Retractions," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1153-1160. doi: 10.4236/am.2012.310169.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. E. El-Ahmady, “The Variation of the Density Functions on Chaotic Spheres in Chaotic Space-Like Minkowski Space Time,” Chaos, Solitons and Fractals, Vol. 31, No. 5, 2007, pp. 1272-1278. Hdoi:10.1016/j.chaos.2005.10.112
[2] A. E. El-Ahmady, “Folding of Fuzzy Hypertori and Their Retractions,” Proc. Math. Phys. Soc. Egypt, Vol. 85, No. 1, 2007, pp. 1-10.
[3] A. E. El-Ahmady, “Limits of Fuzzy Retractions of Fuzzy Hyperspheres and Their Foldings,” Tamkang Journal of Mathematics, Vol. 37, No. 1, 2006, pp. 47-55.
[4] A. E. El-Ahmady, “Fuzzy Folding of Fuzzy Horocycle,” Circolo Matematico di Palermo Serie II, Tomo L III, 2004, pp. 443-450. Hdoi:10.1007/BF02875737
[5] A. E. El-Ahmady, “Fuzzy Lobachevskian Space and Its Folding,” The Journal of Fuzzy Mathematics, Vol. 12, No. 2, 2004, pp. 609-614.
[6] A. E. El-Ahmady, “The Deformation Retract and Topological Folding of Buchdahi Space,” Periodica Mathematica Hungarica, Vol. 28, No. 1, 1994, pp. 19-30. Hdoi:10.1007/BF01876366
[7] A. E. El-Ahmady and H. Rafat, “Retraction of Chaotic Ricci Space,” Chaos, Solutions and Fractals, Vol. 41, 2009, pp. 394-400. Hdoi:10.1016/j.chaos.2008.01.010
[8] A. E. El-Ahmady and H. Rafat, “A Calculation of Geodesics in Chaotic Flat Space and Its Folding,” Chaos, Solutions and Fractals, Vol. 30, 2006, pp. 836-844. Hdoi:10.1016/j.chaos.2005.05.033
[9] A. E. El-Ahmady and H. M. Shamara, “Fuzzy Deformation Retract of Fuzzy Horospheres,” Indian Journal of Pure and Applied Mathematics, Vol. 32, No. 10, 2001, pp. 1501-1506.
[10] A. E. El-Ahmady and A. El-Araby, “On Fuzzy Spheres in Fuzzy Minkowski Space,” Nuovo Cimento, Vol. 125B, 2010.
[11] A. E. El-Ahmady and A. S. Al-Luhaybi, “Retractions of Spatially Curved Robertson-Walker Space,” The Journal of American Sciences, Vol. 8, No. 5, 2012, pp. 548-553.
[12] A. E. El-Ahmady and A. S. Al-Luhaybi, “A Calculation of Geodesics in Flat Robertson-Walker Space and Its Folding,” International Journal of Applied Mathematics and Statistics, Vol. 32, No. 3, 2013, pp. 82-91.
[13] A. E. El-Ahmady, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mthematics, Vol. 19, No. 4, 2011, pp. 833-838.
[14] M. Arkowitz, “Introduction to Homotopy Theory,” Springer-Village, New York, 2011.
[15] T. Banchoff and S. Lovett, “Differential Geometry of Curves and Surfaces,” India, 2010.
[16] B. A. Dubrovin, A. T. Fomenoko and S. P. Novikov, “Modern Geometry-Methods and Applications,” SpringerVerlage, New York, Heidelberg, Berlin, 1984.
[17] W. Kuhnel, “Differential Geometry Curves—SurfacesManifolds,” American Mathematical Society, 2006.
[18] S. Montiel and A. Ros, “Curves and Surfaces,” American Mathematical Society, Madrid, 2009.
[19] E. D. Demainel, “Folding and Unfolding,” Ph. D. Thesis, Waterloo University, Waterloo, 2001.
[20] A. V. Pogorelov, “Differential Geometry,” Noordhoff, Groningen, 1959.
[21] M. S. El Naschie, “Stress, Stability and Chaos in Structural Engineering,” McGraw-Hill, New York, 1990.
[22] G. L. Naber, “Topology, Geometry and Gauge Fields,” Springer, New York, 2011.
[23] P. l. Shick, “Topology: Point-Set and Geometry,” New York, Wiley, 2007. Hdoi:10.1002/9781118031582
[24] J. Strom, “Modern Classical Homotopy Theory,” American Mathematical Society, 2011.
[25] J. B. Hartle, “Gravity, an Introduction to Einstein’s General Relativity,” Addison-Wesley, New York, 2003.
[26] N. Straumann, “General Relativity with Application to Astrophysics,” Springer-Verlage, New York, 2004.

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