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On the recovery of a curve isometrically immersed in $$\mathbb E^n$$. (English) Zbl 1080.53003
The $$n-1$$ curvature functions of a curve in Euclidean $$n$$-space are assumed to belong to suitable Sobolev spaces, the first $$n-2$$ to have positive values. The derivatives in the Frenet system are understood in the sense of distributions. For given curvature functions existence and uniqueness, up to motions, are established. Moreover, the mapping which associates the curve to given curvature functions is shown to be infinitely differentiable. An example in 3-space shows that uniqueness is not given if the curvature vanishes at some point. More about this case can be found in K. Nomizu, On Frenet equations for curves of class $$C^\infty$$ [Tohoku Math. J., II. Ser. 11, 106–112 (1959; Zbl 0107.15304)], see also W. Fenchel, On the differential geometry of closed space curves [Bull. Am. Math. Soc. 57, 44–54 (1951; Zbl 0042.40006)].

MSC:
 53A04 Curves in Euclidean and related spaces 74K05 Strings
Keywords:
Frenet equations; weak derivatives
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