Abstract
In this paper, we define generalized Sabban frames of curves in \(S^{n}\) and investigate the singularities of the spherical duals of the curves by using invariants with respect to such frames.
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Arnol’d, V.I., Gsein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuzer, Basel (1986)
Arnold, V.I.: The geometry of spherical curves and the algebra of quaternions. Russ. Math. Surv. 50, 1–68 (1995)
Blair, D.E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics, vol. 509. Springer, Berlin (1976)
Bruce, J.W., Giblin, P.J.: Curves and Singularities, 2nd edn. Cambridge University press, Cambridge (1992)
Damon, J.: The unfolding and determinacy theorems for subgroups of \({\cal{A}}\) and \({\cal{K}}\). Mem. Am. Math. Soc. 50, 306 (1984)
Encheva, R.P., Georgiev, G.H.: Similar Frenet Curves. Result Math. 55, 359–372 (2009)
Griffiths, P.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)
Koenderink, J.J.: Solid Shape. MIT Press, Cambridge (1990)
Porteous, I.R.: Geometric Differentiation for the Intelligence of Curves and Surfaces, 2nd edn. Cambridge University Press, Cambridge (2001)
Porteous, I.R.: Some remarks on duality in \(S^3\). Banach Center Publ. vol. 50, Polish Acad. Sci. Warsaw, pp. 217–226 (2004)
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Izumiya, S., Nagai, T. Generalized Sabban Curves in the Euclidean \({{\varvec{n}}}\)-Sphere and Spherical Duality. Results Math 72, 401–417 (2017). https://doi.org/10.1007/s00025-017-0685-5
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DOI: https://doi.org/10.1007/s00025-017-0685-5