Curves in Lorentzian spaces

E. Nešović; M. Petrović-Torgašev; L. Verstraelen

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 3, page 685-696
  • ISSN: 0392-4041

Abstract

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The notion of ``hyperbolic'' angle between any two time-like directions in the Lorentzian plane L 2 was properly defined and studied by Birman and Nomizu [1,2]. In this article, we define the notion of hyperbolic angle between any two non-null directions in L 2 and we define a measure on the set of these hyperbolic angles. As an application, we extend Scofield's work on the Euclidean curves of constant precession [9] to the Lorentzian setting, thus expliciting space-like curves in L 3 whose natural equations express their curvature and torsion as elementary eigenfunctions of their Laplacian.

How to cite

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Nešović, E., Petrović-Torgašev, M., and Verstraelen, L.. "Curves in Lorentzian spaces." Bollettino dell'Unione Matematica Italiana 8-B.3 (2005): 685-696. <http://eudml.org/doc/196038>.

@article{Nešović2005,
abstract = {The notion of ``hyperbolic'' angle between any two time-like directions in the Lorentzian plane $L^\{2\}$ was properly defined and studied by Birman and Nomizu [1,2]. In this article, we define the notion of hyperbolic angle between any two non-null directions in $L^\{2\}$ and we define a measure on the set of these hyperbolic angles. As an application, we extend Scofield's work on the Euclidean curves of constant precession [9] to the Lorentzian setting, thus expliciting space-like curves in $L^\{3\}$ whose natural equations express their curvature and torsion as elementary eigenfunctions of their Laplacian.},
author = {Nešović, E., Petrović-Torgašev, M., Verstraelen, L.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {685-696},
publisher = {Unione Matematica Italiana},
title = {Curves in Lorentzian spaces},
url = {http://eudml.org/doc/196038},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Nešović, E.
AU - Petrović-Torgašev, M.
AU - Verstraelen, L.
TI - Curves in Lorentzian spaces
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/10//
PB - Unione Matematica Italiana
VL - 8-B
IS - 3
SP - 685
EP - 696
AB - The notion of ``hyperbolic'' angle between any two time-like directions in the Lorentzian plane $L^{2}$ was properly defined and studied by Birman and Nomizu [1,2]. In this article, we define the notion of hyperbolic angle between any two non-null directions in $L^{2}$ and we define a measure on the set of these hyperbolic angles. As an application, we extend Scofield's work on the Euclidean curves of constant precession [9] to the Lorentzian setting, thus expliciting space-like curves in $L^{3}$ whose natural equations express their curvature and torsion as elementary eigenfunctions of their Laplacian.
LA - eng
UR - http://eudml.org/doc/196038
ER -

References

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  1. BIRMAN, G. S. - NOMIZU, K., Trigonometry in Lorentzian geometry, Amer. Math. Monthly91, (1984), 543-549. Zbl0555.51009MR764793
  2. BIRMAN, G. S. - NOMIZU, K., The Gauss-Bonnet theorem for 2-dimensional space-times, Michigan Math. J., 31 (1984), 77-81. Zbl0591.53053MR736471
  3. BORSUK, K. - SZMIELEW, W., Foundations of Geometry, North-Holland (Amsterdam, 1960). Zbl0093.33301
  4. CHEN, B. Y., Total mean curvature and submanifolds of finite type, World Scientific (Singapore, 1984). Zbl0537.53049MR749575
  5. F. DILLEN - L. VERSTRAELEN (eds.), Handbook of Differential geometry, Vol. I (Elsevier, Amsterdam, 2000). Zbl1069.00010MR1736851
  6. NEŠOVIĆ, E., Differential geometry of curves in Minkowski space, Doctoral thesis, University of Kragujevac, Faculty of Science (Kragujevac, 2002). 
  7. O'NEILL, B., Semi-Riemannian Geometry, Academic Press (New York, 1983). Zbl0531.53051MR719023
  8. PETROVIĆ, M. - VERSTRAELEN, J. - VERSTRAELEN, L., Principal normal spectral variations of space curves, Proyecciones19 (2000), 141-155. MR1778799
  9. SCOFIELD, P. D., Curves of constant precession, Amer. Math. Monthly, 102 (1995), 531-537. Zbl0881.53002MR1336639
  10. STRUIK, D. J., Lectures on classical differential geometry, Addison-Wesley (Boston, 1950). Zbl0105.14707MR36551
  11. SYNGE, J. L., Relativity: the special theory, North-Holland (Amsterdam, 1972). Zbl0071.21804

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