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The total squared curvature of closed curves. (English) Zbl 0554.53013
The authors study the total squared curvature \(\oint k^ 2 ds\) of closed curves in spaces of constant curvature. The critical points of this functional are called closed free elastica. In the case of curves in a zero-dimensional space form a complete classification of such elastica is given. For arbitrary closed curves in the hyperbolic plane the authors obtain the following inequality: \(\oint k^ 2 ds\geq 4\pi\).
Reviewer: U.Pinkall

MSC:
53A35 Non-Euclidean differential geometry
53A04 Curves in Euclidean and related spaces
58E99 Variational problems in infinite-dimensional spaces
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