Djavvat KHADJIEV, İdris ÖREN, Ömer PEKŞEN

Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry}

Let M(n, p) be the group of all motions of an n-dimensional pseudo-Euclidean space of index p. It is proved that the complete system of M(n,p)-invariant differential rational functions of a path (curve) is a generating system of the differential field of all M(n,p)-invariant differential rational functions of a path (curve), respectively. A fundamental system of relations between elements of the complete system of M(n,p)-invariant differential rational functions of a path (curve) is described.

Anahtar Kelimeler:

Key words: Curve, differential invariant, pseudo-Euclidean geometry, Minkowski geometry

Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry}

Keywords:

Key words: Curve, differential invariant, pseudo-Euclidean geometry, Minkowski geometry,

PDF

___

Aslaner, R. and Boran, A. I.: On the geometry of null curves in the Minkowski 4-space. Turkish J. of Math. 32, 1–8 (2008).
Bejancu, A.: Lightlike curves in Lorentz manifolds. Publ. Math. Debrecen. 44, 145–155 (1994).
B´ erard B. L. and Charuel X.: A generalization of Frenet’s frame for nondegenerate quadratic forms with any index. In: S´ eminaire de th´ eorie spectrale et g´ eom´ etrie. Ann´ ee 2001–2002, St. Martin d’H´ eres: Universit´ e de Grenoble I, Institut Fourier, S´ emin. Th´ eor. Spectr. G´ eom. 20, 101–130 (2002).
Bini, D., Geralico, A. and Jantzen, R. T.: Frenet-Serret formalism for null world lines. Class. Quantum Grav. 23, 3963–3981 (2006).
Bonnor, W.: Null curves in a Minkowski spacetime. Tensor, N. S. 20, 229–242 (1969).
Borisov Yu. F.: Relaxing the a priori constraints of the fundamental theorem of space curves in E n l . Siberian Math J. 38, No. 3, 411–427 (1997).
Borisov Yu. F.: On the theorem of natural equations of a curve. Siberian Math. J. 40, No. 4, 617–621 (1999). C ¸ ¨ oken, C. and C ¸ ift¸ci, ¨ U.: On the Cartan curvatures of a null curve in Minkowski spacetime. Geometriae Dedicata. 114, 71–78 (2005).
Duggal, K. L. and Becancu A.: Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications. Dordrecht, Boston, London. Kluwer Acad. Publ. 1996.
Ferr´ andez, A., Gim´ enez, A. and Lucas, P.: Degenerate curves in pseudo-Euclidean spaces of index two. In: Mladenov, Ivailo M. (ed.) et al., Proceedings of the 3rd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 14–23, 2001. Soﬁa: Coral Press Scientiﬁc Publishing. 209–223 (2002).
Ferr´ andez, A., Gim´ enez, A. and Lucas, P.: s -degenerate curves in Lorentzian space forms. J. Geom. Phys. 45, No. 1–2, 116–129 (2003).
Formiga, L. B. and Romero, C.: On the diﬀerential geometry of time-like curves in Minkowski spacetime. Am. J. Phys. 74(10), 1012–1016 (2006).
Ichimura, H.: Time-like and space-like curves in Frenet-Serret formalisms. Thesis. Hadronic J. Suppl. 3, No. 1, 1–94 (1987).
Inoguchi, J. and Lee, S.: Null curves in Minkowski 3-space, International Electronic Journal of Geometry 1 No. 2, 40–83 (2008).
Kaplansky, I.: An Introduction to Diﬀerential Algebra. Paris. Hermann 1957.
Khadjiev, D.: An Application of Invariant Theory to Diﬀerential Geometry of Curves, Fan Publ., Tashkent, 1988. [Russian]
Khadjiev, D. and Pek¸sen ¨ O.: The complete system of global integral and diﬀerential invariants for equi-aﬃne curves. Diﬀerential Geometry and its Applications. 20, 167–175 (2004).
Pek¸sen, ¨ O., Khadjiev, D. and ¨ Oren, I: Invariant parametrizations and complete systems of global invariants of curves in the pseudo-euclidean geometry, Turk. J. Math., 35 1–14 (2011).(in Press).
Petrovi´ c-Torga˘sev, Ilarslan K. and Ne˘sovi´ c E.: On partially null and pseudo-null curves in the semi-euclidean space R 4 2 . J. Geom. 84, 106–116 (2005)
Urbantke H.: Local diﬀerential geometry of null curves in conformally ﬂat space-time. J. Math. Phys. 30(10), 2238–2245 (1989).
Weyl, H.: The Classical Groups. Their Invariants and Representations. Princeton-New Jersey. Princeton Univ. Press 1946.
Yilmaz, S. and Turgut, M.: On the diﬀerential geometry of curves in Minkowski space-time I. Int. J. Contemp. Math. Sciences. 3, No. 27, 1343–1349 (2008).