JOURNAL BROWSE
Search
Advanced SearchSearch Tips
BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
BERTRAND CURVES AND RAZZABONI SURFACES IN MINKOWSKI 3-SPACE
Xu, Chuanyou; Cao, Xifang; Zhu, Peng;
  PDF(new window)
 Abstract
In this paper, we generalize some results about Bertrand curves and Razzaboni surfaces in Euclidean 3-space to the case that the ambient space is Minkowski 3-space. Our discussion is divided into three different cases, i.e., the parent Bertrand curve being timelike, spacelike with timelike principal normal, and spacelike with spacelike principal normal. For each case, first we show that Razzaboni surfaces and their mates are related by a reciprocal transformation; then we give Bcklund transformations for Bertrand curves and for Razzaboni surfaces; finally we prove that the reciprocal and Bcklund transformations on Razzaboni surfaces commute.
 Keywords
Bertrand curve;Razzaboni surface;Minkowski 3-space;reciprocal transformation;Bcklund transformation;
 Language
English
 Cited by
1.
ON TIMELIKE BERTRAND CURVES IN MINKOWSKI 3-SPACE,;;

호남수학학술지, 2016. vol.38. 3, pp.467-477 crossref(new window)
1.
ON TIMELIKE BERTRAND CURVES IN MINKOWSKI 3-SPACE, Honam Mathematical Journal, 2016, 38, 3, 467  crossref(new windwow)
 References
1.
H. Balgetir, M. Bektas, and M. Ergot, Bertrand curves for nonnull curves in 3- dimensional Lorentzian space, Hadronic J. 27 (2004), no. 2, 229-236.

2.
J. F. Burke, Bertrand curves associated with a pair of curves, Math. Maga. 34 (1960), no. 1, 60-62. crossref(new window)

3.
N. Ekmekci and K. Ilarslan, On Bertrand curves and their characterization, Differ. Geom. Dyn. Syst. 3 (2001), no. 2, 17-24.

4.
S. Izumiya and N. Takeuchi, Generic properties of helices and Bertrand curves, J. Geom. 74 (2002), no. 1-2, 97-109. crossref(new window)

5.
S. Izumiya and A. Takiyama, A time-like surface in Minkowski 33-space which contains pseudocircles, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 1, 127-136.

6.
M. Kulahci and M. Ergut, Bertrand curves of AW(k)-type in Lorentzian space, Nonlinear Anal. 70 (2009), no. 4, 1725-1731. crossref(new window)

7.
R. Lopez, Differential geometry of curves and surfaces in Lorent-Minkowski spaces, arXiv:0810.3351vl[math.DG]

8.
B. O'Nell, Semi-Riemannian Geometry, Academic Press, 1983.

9.
A. Razzaboni, Un teorema del signor Demartres generalizzato, Ven. Ist. Atti. 60 (1901), 757-768.

10.
A. Razzaboni, Delle superficie nelle quali un sistema di geodetiche sono curve del Bertrand, Boligna Mem. 10 (1903), 539-548.

11.
W. K. Schief, On the integrability of Bertrand curves and Razzaboni surfaces, J. Geom. Phys. 45 (2003), no. 1-2, 130-150. crossref(new window)

12.
M. Yilmaz and M. Bektas, General properties of Bertrand curves in Riemann-Otsuki space, Nonlinear Anal. 69 (2008), no. 10, 3225-3231. crossref(new window)