References
H. W. Guggenheimer, Differential Geometry (McGraw-Hill Book Company, Inc., New York, 1963).
Yu. A. Aminov, Differential Geometry and Topology of Curves (Nauka, Moscow, 1987).
W. Klingenberg, A Course in Differential Geometry (Springer-Verlag, New York, 1978).
F. Nevanlinna and R. Nevanlinna, Absolute Analysis (Springer-Verlag, Berlin, 1959).
M. Spivak, A Comprehensive Introduction to Differential Geometry (Publ. of Perish, Inc., Berkeley, CA, 1979), Vol. 2.
R. G. Aripov, “Equivalence of Paths in an n-Dimensional Complex Vector Space under the Action of the Group SO(n, R),” Dokl. Akad. Nauk UzSSR 6, 8–10 (1986).
R. G. Aripov, “A Criterion of G-Equivalence of Finite Systems of Regular Paths in C n for Real Orthogonal Groups G,” Uzb. Matem. Zhurn. Nos. 5–6, 17–22 (1992).
R. G. Aripov, “Algebraic Structure of a Differential Field of G-Invariant Differential Rational Functions of Paths in the Space C n for Complex Orthogonal Groups G,” Uzb. Matem. Zhurn. No. 1, 3–11 (1997).
R. G. Aripov, “A Theorem on Existence and Uniqueness of a System of Regular Paths for the Action of Complex Orthogonal Groups in C n,” Uzb. Matem. Zhurn. No. 2, 6–12 (2001).
S. S. Chern, “Curves and Surfaces in Euclidean Space,” Global Diff. Geom. 27, 99–139 (1989).
F. Farris, “Classifying Curves in Space and Space-Time,” Exp. Math. 6, 81–89 (1988).
D. Khadzhiev, “On an Invariant Parameter for Curves,” Dokl. Akad. Nauk UzSSR. No. 7, 5 (1986).
M. Kosters, “Curvature-Dependent Parametrization of Curves and Surfaces,” Comput. Sci. Notes, Univ. Groningen, Dep. Math. and Comput. Sci. 8915, 1–20 (1989).
G. S. Molnar, “On Some Questions Concerning the Differential Geometry of Curves in n-Dimensional Euclidean Spaces,” Publ. Math. 30(1–2), 57–73 (1983).
B. Wegner, “Some New Developments in the Global Theory of Curves,” in Proceedings of the 24th national conference of geometry and topology, Timisoara, Romania, July 5–9, 1994. Part 1. Lectures Ed. by Albu Adrian C. et al. (Editora Mirton, Timisoara, 1996), pp. 275–283.
J. L. Weiner, “An Inequality Involving the Length, Curvature and Torsions of a Curve in Euclidean n-Space,” Pacific J. Math. 74(2), 531–534 (1978).
D. Khadzhiev, Application of the Theory of Invariants to the Differential Geometry of Curves (Tashkent, 1988) [in Russian].
D. Khadzhiev and O. Peksen, “The Complete System of Global Differential and Integral Invariants of Equiaffine Curves,” Diff. Geom. And Appl. 20, 168–175 (2004).
O. Peksen and D. Khadzhiev, “On Invariants of Curves in Centro-Affine Geometry,” J. Math. Kyoto. Univ. (JMKYAZ) 44(3), 603–613 (2004).
A. D. Alexandrov and Yu. G. Reshetnyak, General Theory of Irregular Curves (Kluwer Acad. Publ., 1989).
E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (Univ. of Toronto Press, 1968).
I. Kaplansky, Introduction to Differential Algebra (Hermann & Cie, Publ. Inst. Math. Univ. Nancago. V. Paris, 1957; Inost. Lit., Moscow, 1959).
Y. A. Aminov, The Geometry of Submanifolds (Gordon and Breach Sciences Publishers, London-Amsterdam, 2001).
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Original Russian Text © R.G. Aripov, D. Khadzhiev, 2007, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2007, No. 7, pp. 3–16.
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Aripov, R.G., Khadzhiev, D. The complete system of global differential and integral invariants of a curve in Euclidean geometry. Russ Math. 51, 1–14 (2007). https://doi.org/10.3103/S1066369X07070018
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DOI: https://doi.org/10.3103/S1066369X07070018