On Elastic Klein Bottle and Fundamental Groups
Author(s)
A. E. El-Ahmady*
ABSTRACT
The purpose of this paper is to give a combinatorial characterization and also construct representations of the fundamental group of the submanifolds of elastic Klein Bottle by using some geometrical transformations. The homotopy groups of the limit elastic Klein Bottle are presented. The fundamental groups of some types of geodesics in elastic Klein Bottle are discussed. New types of homotopy maps are deduced. Theorems governing this connection are achieved.
Cite this paper
A. El-Ahmady, "On Elastic Klein Bottle and Fundamental Groups," Applied Mathematics, Vol. 4 No. 3, 2013, pp. 499-504. doi: 10.4236/am.2013.43074.
A. El-Ahmady, "On Elastic Klein Bottle and Fundamental Groups," Applied Mathematics, Vol. 4 No. 3, 2013, pp. 499-504. doi: 10.4236/am.2013.43074.
References
[1] A. E. El-Ahmady, “Folding and Fundamential Groups of Buchdahi Space,” Indian Journal of Science and Technology, Vol. 6, No. 1, 2013, pp. 3940-3945. [2] A. E. El-Ahmady, “On the Fundamential Group and Folding of Klein Bottle,” International Journal of Applied Mathematics and Statistics, Vol. 37, No. 6, 2013, pp. 56-64. [3] A. E. El-Ahmady, “The Geodesic Deformation Retract of Klein Bottle and Its Folding,” International Journal of Nonlinear Science, Vol. 9, No. 3, 2011, pp. 1-8. [4] A. E. El-Ahmady, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mathematics, Vol. 19, No. 4, 2011, pp. 833-838. [5] A. E. El-Ahmady, “Folding of Fuzzy Hypertori and Their Retractions,” Proceedings of the Mathematical and Physical Society of Egypt, Vol. 85, No. 1, 2007, pp. 1-10. [6] 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. [7] 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 [8] 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 [9] A. E. El-Ahmady, “Fuzzy Lobachevskian Space and Its Folding,” The Journal of Fuzzy Mathematics, Vol. 12, No. 2, 2004, pp. 609-614. [10] A. E. El-Ahmady, “The Deformation Retract and Topological Folding of Buchdahi Space,” Periodica Mathematica Hungarica, Vol. 28, No. 1, 1994, pp. 19-30. doi:10.1007/BF01876366 [11] A. E. El-Ahmady, “Folding of Some Types of Einstein Spaces,” The International Journal of Nonlinear Science, in Press. [12] A. E. El-Ahmady and K. Al-Onema, “On the Folding of Lobachevsky Space,” International Journal of Applied Mathematics and Statistics, Vol. 4, No. 10, 2013, pp. 13-23. [13] A. E. El-Ahmady, “The Deformation Retract and Topological Folding of the Schwarzchild Space,” Journal of Mathematical Physics, Vol. 25, No. 3, 1991. [14] A. E. El-Ahmady and N. Al-Hazmi, “Retractions of One Dimensional Manifold,” Applied Mathematics, Vol. 3, No. 10, 2012, pp. 1135-1143. [15] A. E. El-Ahmady and E. Al-Hesiny, “Folding and Deformation Retract of Hyperhelix,” Journal of Mathematics and Statistics, Vol. 8, No. 2, 2012, pp. 241-247. [16] A. E. El-Ahmady and E. Al-Hesiny, “The Topological Folding of the Hyperbola in Minkowski 3-Space,” The International Journal of Nonlinear Science, Vol. 11, No. 4, 2011, pp. 451-458. [17] 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. 33, No. 3, 2013, pp. 83-91. [18] A. E. El-Ahmady and A. Al-Rdade, “A Geometrical Characterizations of Reissner-Nordstr?m Space Time and Its Retractions,” International Journal of Applied Mathematics and Statistics, Vol. 33, No. 3, 2013, pp. 83-91. [19] A. E. El-Ahmady and A. S. Al-Luhaybi, “Ageometrical Characterization of Spatially Curved Robertson-Walker Space and Its Retractions,” Applied Mathematics, Vol. 3, No. 10, 2012, pp. 1153-1160. [20] M. Reid and B. Szendroi, “Topology and Geometry,” Cambridge University Press, Cambridge, New York, 2005. [21] M. Arkowitz, “Introduction to Homotopy Theory,” Springer-Verlage, New York, 2011. doi:10.1007/978-1-4419-7329-0 [22] P. I. Shick, “Topology: Point-Set and Geometry,” Wiley, New York, 2007. [23] J. Strom, “Modern Classical Homotopy Theory,” American Mathematical Society, 2011.
[1] A. E. El-Ahmady, “Folding and Fundamential Groups of Buchdahi Space,” Indian Journal of Science and Technology, Vol. 6, No. 1, 2013, pp. 3940-3945. [2] A. E. El-Ahmady, “On the Fundamential Group and Folding of Klein Bottle,” International Journal of Applied Mathematics and Statistics, Vol. 37, No. 6, 2013, pp. 56-64. [3] A. E. El-Ahmady, “The Geodesic Deformation Retract of Klein Bottle and Its Folding,” International Journal of Nonlinear Science, Vol. 9, No. 3, 2011, pp. 1-8. [4] A. E. El-Ahmady, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mathematics, Vol. 19, No. 4, 2011, pp. 833-838. [5] A. E. El-Ahmady, “Folding of Fuzzy Hypertori and Their Retractions,” Proceedings of the Mathematical and Physical Society of Egypt, Vol. 85, No. 1, 2007, pp. 1-10. [6] 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. [7] 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 [8] 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 [9] A. E. El-Ahmady, “Fuzzy Lobachevskian Space and Its Folding,” The Journal of Fuzzy Mathematics, Vol. 12, No. 2, 2004, pp. 609-614. [10] A. E. El-Ahmady, “The Deformation Retract and Topological Folding of Buchdahi Space,” Periodica Mathematica Hungarica, Vol. 28, No. 1, 1994, pp. 19-30. doi:10.1007/BF01876366 [11] A. E. El-Ahmady, “Folding of Some Types of Einstein Spaces,” The International Journal of Nonlinear Science, in Press. [12] A. E. El-Ahmady and K. Al-Onema, “On the Folding of Lobachevsky Space,” International Journal of Applied Mathematics and Statistics, Vol. 4, No. 10, 2013, pp. 13-23. [13] A. E. El-Ahmady, “The Deformation Retract and Topological Folding of the Schwarzchild Space,” Journal of Mathematical Physics, Vol. 25, No. 3, 1991. [14] A. E. El-Ahmady and N. Al-Hazmi, “Retractions of One Dimensional Manifold,” Applied Mathematics, Vol. 3, No. 10, 2012, pp. 1135-1143. [15] A. E. El-Ahmady and E. Al-Hesiny, “Folding and Deformation Retract of Hyperhelix,” Journal of Mathematics and Statistics, Vol. 8, No. 2, 2012, pp. 241-247. [16] A. E. El-Ahmady and E. Al-Hesiny, “The Topological Folding of the Hyperbola in Minkowski 3-Space,” The International Journal of Nonlinear Science, Vol. 11, No. 4, 2011, pp. 451-458. [17] 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. 33, No. 3, 2013, pp. 83-91. [18] A. E. El-Ahmady and A. Al-Rdade, “A Geometrical Characterizations of Reissner-Nordstr?m Space Time and Its Retractions,” International Journal of Applied Mathematics and Statistics, Vol. 33, No. 3, 2013, pp. 83-91. [19] A. E. El-Ahmady and A. S. Al-Luhaybi, “Ageometrical Characterization of Spatially Curved Robertson-Walker Space and Its Retractions,” Applied Mathematics, Vol. 3, No. 10, 2012, pp. 1153-1160. [20] M. Reid and B. Szendroi, “Topology and Geometry,” Cambridge University Press, Cambridge, New York, 2005. [21] M. Arkowitz, “Introduction to Homotopy Theory,” Springer-Verlage, New York, 2011. doi:10.1007/978-1-4419-7329-0 [22] P. I. Shick, “Topology: Point-Set and Geometry,” Wiley, New York, 2007. [23] J. Strom, “Modern Classical Homotopy Theory,” American Mathematical Society, 2011.