Geometriae Dedicata

, Volume 167, Issue 1, pp 239–244 | Cite as

Approximations of periodic functions to \(\mathbb R ^n\) by curvatures of closed curves

Original Paper


We show that for any \(n\) real periodic functions \(f_1,\ldots , f_n\) with the same period, such that \(f_i>0\) for \(i<n\), and a real number \(\varepsilon >0\), there is a closed curve in \(\mathbb R ^{n+1}\) with curvatures \(\kappa _1, \ldots , \kappa _n\) such that \(\left| \kappa _{i(t)}-f_{i(t)}\right|<\varepsilon \) for all \(i\) and \(t\). This does not hold for parametric families of closed curves in \(\mathbb R ^{n+1}\).


Curvatures Frenet frame \(h\)-principle 

Mathematics Subject Classification (2000)

Primary: 530A4 Secondary: 53C21 53C42 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.CINVESTAVMexico, D.F.Mexico

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