Some Characterizations of Constant Breadth Spacelike Curves in Minkowski 4-space

Abstract

In this study, the differential equation characterizations of constant breadth spacelike curves are given in the Minkowski4-space E4. Furthermore, a criterion for a spacelike curve to be a curve of constant breadth in Eis introduced. As an example, theobtained results are applied to the case that the curvatures k, k, k3are constants and are discussed

Keywords

• Constant breadth curve,spacelike curve; Frenet frame

Huseyin Kocayi˘git and Zennure C¸ ic¸ek

Öz

References

1. Ball, N. H., On ovals, American Mathematical Monthly, 37(7) (1930), 348-353.
2. Barbier, E., Note sur le probl´eme de I’aiguille et le jeu du joint couvert., Journal de Math´ematiques Pures et Appliqu´ees, 2(5) (1860), 273-286.
3. Blaschke, W., Konvexe bereichee gegebener konstanter breite und kleinsten inhalt, Math. Annalen, B. 76 (1915), 504-513.
4. Blaschke, W., Einige Bemerkungen ¨uber Kurven und Fl¨achen konstanter Breite. Ber. Verh. s¨achs. Akad. Leipzig, 67 (1915), 290-2
5. Chung, H.C., A Differential-Geometric Criterion for a Space Curve to be Closed, Proc. Amer. Math. Soc. 83(2) (1981), 357-361.
6. Euler, L., De curvis triangularibus, Acta Acad. Petropol., 3-30, 1778 (1780).
7. Fujivara, M., On space curves of constant breadth, Tohoku Mathematical Journal, 5 (1914), 180-184.
8. Kazaz, M., ¨Onder, M., Kocayiˇgit H., Spacelike curves of constant breadth in Minkowski 4-space, Int. Journal of Math. Analysis, 2(22) (2008), 1061 – 1068.

9. Kocayiˇgit, H., ¨Onder, M., Spacelike curves of constant breadth in Minkowski 3-space, Annali di Matematica, 192(5) (2013), 805-8
10. K¨ose, ¨O., On space curves of constant breadth, Do˘ga Tr. J. Math., 10(1) (1986), 11-14.
11. Maˇgden, A., K¨ose, ¨O., On the curves of constant breadth in E4space, Tr. J. of Mathematics, 21(3) (1997), 277-284.
12. Mellish, A. P., Notes on differential geometry, Annals of Mathematics, 32(1) (1931), 181-190.
13. ¨Onder, M., Kocayiˇgit, H., Candan, E., Differential equations characterizing timelike and spacelike curves of constant breadth in Minkowski 3-space E 3, J. Korean Math. Soc. 48( 1, J. Korean Math. Soc. 48( ), (2011), 849-866.
14. Reuleaux, F., The Kinematics of Machinery, Translated by A.B.W. Kennedy, Dover Pub. Nex York, 1963.
15. Ross, S.L., Differential Equations, John Wiley and Sons, Inc., New York, (1974), 440-468.
16. Sezer, M., Differential equations characterizing space curves of constant breadth and a criterion for these curves, Turkish J. of Math., 13(2) (1989), 70-78.
17. Smakal, S., Curves of constant breadth, Czechoslovak Mathematical Journal, 23() (1973), 86–94.
18. Struik, D.J., Differential Geometry in the Large, Bulletin Amer. Mathem. Soc., 37 (1931), 49-62.
19. Tanaka, H., Kinematics Design of Com Follower Systems, Doctoral Thesis, Columbia Univ., (1976).
20. Walrave, J., Curves and surfaces in Minkowski space, K. U. Leuven, Faculty of Sciences, Ph.D. Thesis, 1995.