Geodesic
On the curvatures of a curve in $n$-dimensional Euclidean space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2021), pp. 54-66.

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Formulas are found for calculating the curvatures of an implicitly defined curve in $n$-dimensional Euclidean space. For these curves, Beltrami's theorem is generalized, which he proved in the three-dimensional case.
Keywords: smooth curve, curvature, implicit definition of a curve, touching $k$-plane of a curve, Beltrami theorem. 
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A.M. Shelekhov. On the curvatures of a curve in $n$-dimensional Euclidean space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2021), pp. 54-66. http://geodesic.mathdoc.fr/item/IVM_2021_11_a5/

[1] Jordan C., “Sur la téorie des courbes dans l'espace à n dimensions”, Oeuvres, 4, Gauthier-Villars, et Blanchard, Paris, 1964, 337–339

[2] Goldman R., “Curvature formulas for implicit curves and surfaces”, Comput. Aided Geometric Design, 22:7 (2005), 632–658 | DOI | Zbl

[3] Rozenfeld B. A., Mnogomernye prostranstva, Nauka, M., 1966