Retractions of One Dimensional Manifold

Affiliation(s)

Mathematics Department, Faculty of Science, Taibah University, Madinah, Saudi Arabia.

Mathematics Department, Faculty of Science, Taibah University, Madinah, Saudi Arabia.

ABSTRACT

Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional manifold into itself and onto geodesics is deduced. Also, the isometric and topological folding in each case and the relation between the deformations retracts after and before folding has been obtained. New types of conditional folding are described.

Our aim in the present article is to introduce and study types of retraction of one dimensional manifold. New types of geodesics in one dimensional manifold are presented. The deformation retracts of one dimensional manifold into itself and onto geodesics is deduced. Also, the isometric and topological folding in each case and the relation between the deformations retracts after and before folding has been obtained. New types of conditional folding are described.

Cite this paper

A. El-Ahmady and N. Al-Hazmi, "Retractions of One Dimensional Manifold,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1135-1143. doi: 10.4236/am.2012.310167.

A. El-Ahmady and N. Al-Hazmi, "Retractions of One Dimensional Manifold,"

References

[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. doi: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, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mthematics, Vol. 19, No. 4, 2011, pp. 833-838.

[5] A. E. El-Ahmady, “Fuzzy Folding of Fuzzy Horocycle,” Circolo Matematico di Palermo Serie II, Tomo L III, 2004, pp. 443-450. doi:10.1007/BF02875737

[6] A. E. El-Ahmady, “Fuzzy Lobachevskian Space and Its Folding,” The Journal of Fuzzy Mathematics, Vol. 12, No. 2, 2004, pp. 609-614.

[7] 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

[8] 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

[9] 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

[10] 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.

[11] A. E. El-Ahmady and A. El-Araby, “On Fuzzy Spheres in Fuzzy Minkowski Space,” Nuovo Cimento, Vol. 125B, 2010.

[12] M. Arkowitz, “Introduction to Homotopy Theory,” Springer-Village, New York, 2011.

[13] G. L. Naber, “Topology, Geometry and Gauge Fields,” Foundations, Springer, New York, 2011.

[14] G. L. Naber, “Topology, Geometry and Gauge Fields,” Interactions, Springer, New York, 2011.

[15] M. Reid and B. Szendroi, “Topology and Geometry,” Cambridge, New York, 2005. doi:10.1017/CBO9780511807510

[16] P. l. Shick, “Topology: Point-Set and Geometry,” Wiley, New York, 2007. doi:10.1002/9781118031582

[17] J. Strom, “Modern Classical Homotopy Theory,” American Mathematical Society, 2011.

[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. doi: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, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mthematics, Vol. 19, No. 4, 2011, pp. 833-838.

[5] A. E. El-Ahmady, “Fuzzy Folding of Fuzzy Horocycle,” Circolo Matematico di Palermo Serie II, Tomo L III, 2004, pp. 443-450. doi:10.1007/BF02875737

[6] A. E. El-Ahmady, “Fuzzy Lobachevskian Space and Its Folding,” The Journal of Fuzzy Mathematics, Vol. 12, No. 2, 2004, pp. 609-614.

[7] 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

[8] 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

[9] 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

[10] 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.

[11] A. E. El-Ahmady and A. El-Araby, “On Fuzzy Spheres in Fuzzy Minkowski Space,” Nuovo Cimento, Vol. 125B, 2010.

[12] M. Arkowitz, “Introduction to Homotopy Theory,” Springer-Village, New York, 2011.

[13] G. L. Naber, “Topology, Geometry and Gauge Fields,” Foundations, Springer, New York, 2011.

[14] G. L. Naber, “Topology, Geometry and Gauge Fields,” Interactions, Springer, New York, 2011.

[15] M. Reid and B. Szendroi, “Topology and Geometry,” Cambridge, New York, 2005. doi:10.1017/CBO9780511807510

[16] P. l. Shick, “Topology: Point-Set and Geometry,” Wiley, New York, 2007. doi:10.1002/9781118031582

[17] J. Strom, “Modern Classical Homotopy Theory,” American Mathematical Society, 2011.