On the Geometry of Curves in Minkowski 3-Space and Its Foldings

ABSTRACT

We will introduce a new connection between some transformations and some aspects of differential geometry of some curves in Minkowski space. The concept of folding, retractions and contraction on some curves in Minkowski space will be characterized by using some aspects of differential geometry. Types of the deformation retracts of some curves in Minkowski 3-space are obtained. The relations between the foldings and the deformation retracts of some curves are deduced. The connections between some transformations and time like, space like, light like of some curves in Minkowski 3-space are also presented.

Cite this paper

A. El-Ahmady and E. Al-Hesiny, "On the Geometry of Curves in Minkowski 3-Space and Its Foldings,"*Applied Mathematics*, Vol. 4 No. 5, 2013, pp. 746-752. doi: 10.4236/am.2013.45103.

A. El-Ahmady and E. Al-Hesiny, "On the Geometry of Curves in Minkowski 3-Space and Its Foldings,"

References

[1] A. E. El-Ahmady and E. Al-Hesiny, “Conditional Retraction of Some Curves in Minkowski 3-Space,” International Journal of Applied Mathematics and Statistics, Vol. 32, No. 2, 2012, pp. 39-47.

[2] A. E. El-Ahmady, “The Variation of the Density 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

[3] 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.

[4] A. E. El-Ahmady and E. Al-Hesiny, “Folding and Differential Equations of Some Curves in Minkowski Space,” Life Science Journal, Vol. 9, No. 2, 2012, pp. 579-584.

[5] 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. doi:10.3844/jmssp.2012.241.247

[6] A. E. El-Ahmady, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mathematics, Vol. 19, No. 4, 2011, pp. 833-838.

[7] 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.

[8] A. E. El-Ahmady, “The Deformation Retract and Topo Logical Folding of Buchdahi Space,” Periodica Mathematica Hungarica, Vol. 28, No. 1, 1994, pp. 19-30. doi:10.1007/BF01876366

[9] A. E. El-Ahmady, “Folding and Fundamential Group of Buchdahi Space,” Indian Journal of Science and Technology, Vol. 6, No. 1, 2013, pp. 3940-3945.

[10] A. E. El-Ahmady, “On the Fundamental Group and Folding of Klein Bottle,” International Journal of Applied Mathematics and Statistics, Vol. 37, No. 6, 2013, pp. 56-64.

[11] A. E. El-Ahmady, “The Geodesic Deformation Retract of Klein Bottle and Its Folding,” The International Journal of Nonlinear Science, Vol. 12, No. 3, 2011, pp. 323-330.

[12] A. E. El-Ahmady and E. Al-Hesiny, “On Some Curves in Minkowski 3-Space and Its Deformation Retract,” International Journal of Applied Mathematics and Statistics, Vol. 36, No. 6, 2013, pp. 42-53.

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

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

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

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

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

[18] J. M. Lee, “Introduction to Topological Manifolds,” Springer-Verlage, New York, 2011. doi:10.1007/978-1-4419-7940-7

[19] R. Lopez, “Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space,” Instituto de Matematica e Estatistica, University of Sao Paulo, Sao Paulo, 2008.

[1] A. E. El-Ahmady and E. Al-Hesiny, “Conditional Retraction of Some Curves in Minkowski 3-Space,” International Journal of Applied Mathematics and Statistics, Vol. 32, No. 2, 2012, pp. 39-47.

[2] A. E. El-Ahmady, “The Variation of the Density 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

[3] 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.

[4] A. E. El-Ahmady and E. Al-Hesiny, “Folding and Differential Equations of Some Curves in Minkowski Space,” Life Science Journal, Vol. 9, No. 2, 2012, pp. 579-584.

[5] 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. doi:10.3844/jmssp.2012.241.247

[6] A. E. El-Ahmady, “Retraction of Chaotic Black Hole,” The Journal of Fuzzy Mathematics, Vol. 19, No. 4, 2011, pp. 833-838.

[7] 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.

[8] A. E. El-Ahmady, “The Deformation Retract and Topo Logical Folding of Buchdahi Space,” Periodica Mathematica Hungarica, Vol. 28, No. 1, 1994, pp. 19-30. doi:10.1007/BF01876366

[9] A. E. El-Ahmady, “Folding and Fundamential Group of Buchdahi Space,” Indian Journal of Science and Technology, Vol. 6, No. 1, 2013, pp. 3940-3945.

[10] A. E. El-Ahmady, “On the Fundamental Group and Folding of Klein Bottle,” International Journal of Applied Mathematics and Statistics, Vol. 37, No. 6, 2013, pp. 56-64.

[11] A. E. El-Ahmady, “The Geodesic Deformation Retract of Klein Bottle and Its Folding,” The International Journal of Nonlinear Science, Vol. 12, No. 3, 2011, pp. 323-330.

[12] A. E. El-Ahmady and E. Al-Hesiny, “On Some Curves in Minkowski 3-Space and Its Deformation Retract,” International Journal of Applied Mathematics and Statistics, Vol. 36, No. 6, 2013, pp. 42-53.

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

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

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

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

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

[18] J. M. Lee, “Introduction to Topological Manifolds,” Springer-Verlage, New York, 2011. doi:10.1007/978-1-4419-7940-7

[19] R. Lopez, “Differential Geometry of Curves and Surfaces in Lorentz-Minkowski Space,” Instituto de Matematica e Estatistica, University of Sao Paulo, Sao Paulo, 2008.