Complete systems of differential invariants of vector fields in a euclidean space

The system of generators of the differential field of all G-invariant differential rational functions of a vector field in the n-dimensional Euclidean space Rn is described for groups G=M(n) and G=SM(n), where M(n) is the group of all isometries of Rn and SM(n) is the group of all euclidean motions of Rn. Using these results, vector field analogues of the first part of the Bonnet theorem for groups Aff(n), M(n), SM(n) in Rn are obtained, where Aff(n) is the group of all affine transformations of Rn. These analogues are given in terms of the first fundamental form and Christoffel symbols of a vector field.

Anahtar Kelimeler:

Vector field; Christoffel symbol; Bonnet theorem; Differential invariant

Complete systems of differential invariants of vector fields in a euclidean space

Keywords:

Vector field; Christoffel symbol; Bonnet theorem; Differential invariant,

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Aminov, J. A.: Sources of curvature of a vector field. Math. USSR Sbornik 9, No. 2, 210-224 (1969).
Aminov, Yu. A.: The Geometry of Vector Fields. Amsterdam. Gordan and Breach Sciences Publ. 2000.
Aminov, Yu. A.: The Geometry of Submanifolds. Amsterdam. Gordan and Breach Sciences Publ. 2001.
Antoneli, F., Baptistelli, P. H., Dias, A. P. S., Manoel, M.: Invariant theory and reversible-equivariant vector fields. Journal of Pure and Applied Algebra 213, 649-663 (2009).
Aripov, R. G., Khadjiev, D.: The complete system of differential and integral invariants of a curve in Euclidean geometry. Russian Mathematics (Iz VUZ) 51, No. 7, 1-14 (2007).
Baptistelli, P. H., Manoel, M. G.: Some results on reversible -equivariant vector fields. Cadornos De Matema’tica 6, 237-263 (2005).
Bushgens, S.: Geometry of vector fields. Rep. Acad. Sci. USSR, Ser. Mat. 10, 73-96 (1946) [Russian].
Doffou, M. J., Grossman, R. L.: The symbolic computation of differential invariants of polynomial vector field systems using trees. Proceedings of the 1995 International Symposium of Symbolic and Algebraic Computation, A. H. M. Levelt, editor, ACM, 26-31 (1995).
Golubitsky, M., Stewart, I.N., Schaeffer, D.G.: Singularities and Groups in Bifurcation Theory, vol. 2. Applied Mathematical Sciences 69. New York. Springer-Verlag 1985.
Kaplansky, I.: An Introduction to Differential Algebra. Paris. Hermann 1957.
Khadjiev, D.: Application of Invariant Theory to the Differential Geometry of Curves. Tashkent. Fan Publ. 1988. [Russian] (Zbl 0702.53002)
Khadjiev, D., Pek¸sen, O.: The complete system of global integral and differential invariants for equi-affine curves. ¨ Differ. Geom. Appl. 20, 167-175 (2004).
Li, A.-M., Simon, U., Zhao, G.: Global Affine Differential Geometry of Hypersurfaces. Berlin-New York. Walter de Gruyter 1993.
Pek¸sen, O., Khadjiev, D.: On invariants of curves in centro-affine geometry. J. Math. Kyoto Univ. (JMKYAZ) 44-3, ¨ 603-613 (2004).
Sibirskii, K. S.: Algebraic Invariants of Differential Equations and Matrices. Kishinev. Stiintsa 1976 [Russian].
Sibirskii, K. S.: Introduction to the Algebraic Invariants of Differential Equations. New York. Manchester University Press 1988.
Sibirskii, K. S., Taku, V. D.: Algebraic invariants of many-dimensional differential systems. Differentisialnyje Uravnenija 12, 281-291 (1988) [Russian].
Vekua, I. N.: Foundations of Tensor Analysis and Theory of Covariants. Moscow. Nauka 1978 [Russian].
Weyl, H.: The Classical Groups. Their Invariants and Representations. Princeton-New Jersey. Princeton Univ. Press 1946.