Approximations of periodic functions to $$\mathbb R ^n$$ by curvatures of closed curves
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Approximations of periodic functions to Rn by curvatures of closed curves J. Mostovoy · R. Sadykov
Received: 26 May 2012 / Accepted: 4 December 2012 / Published online: 12 December 2012 © Springer Science+Business Media Dordrecht 2012
Abstract We show that for any n real periodic functions f 1 , . . . , f n with the same period, n+1 with such that f i > 0 for i < n, and a real number ε > 0, there is a closed curve in R curvatures κ1 , . . . , κn such that κi(t) − f i(t) < ε for all i and t. This does not hold for parametric families of closed curves in Rn+1 . Keywords
Curvatures · Frenet frame · h-principle
Mathematics Subject Classification (2000)
Primary: 530A4 · Secondary: 53C21 · 53C42
1 Introduction Let α : S 1 → Rn+1 be a closed C n+1 -differentiable curve such that for each t ∈ S 1 the vectors {α (t), α (t), . . . , α n (t)}
(1)
are linearly independent. We shall refer to such a curve as a closed Frenet curve. The GramSchmidt process turns the set of vectors (1) into a set of orthonormal vectors {e1 , . . . , en }, which, together with a unique unit vector en+1 , forms a positively oriented orthonormal basis {e1 , . . . , en+1 } called the Frenet frame [9]. The curvatures κ1α , . . . , κnα of the curve α at the point t are defined by induction by means of the Frenet formulae
J. Mostovoy (B) · R. Sadykov CINVESTAV, Col. San Pedro Zacatenco, CP 07360, Mexico, D.F., Mexico e-mail: [email protected] R. Sadykov e-mail: [email protected]
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Geom Dedicata (2013) 167:239–244
1 e = κ1α e2 , |α | 1 1 α e = −κiα ei + κi+1 ei+2 , for i = 1, . . . , n − 1, |α | i+1 1 = −κnα en . e |α | n+1 α are strictly positive [9, Proposition 1.3.4]. We say It follows that the curvatures κ1α , . . . , κn−1 that the curve α is twisted if the last curvature κnα is nowhere zero. A curvature-like function is a periodic function to (R+ )n−1 × R, that is, a map S 1 → (R+ )n−1 × R. Every closed Frenet curve α gives rise to the curvature-like function
jα : t → (κ1α (t), κ2α (t), . . . , κnα (t)). A curvature-like function of the form jα , for some curve α, is called holonomic. Given a curvature-like function s and a real number ε > 0, we say that a closed curve α is a holonomic ε-approximation of s if jα is ε-close to s, that is, for each t ∈ S 1 we have |s(t) − jα (t)| < ε. Similarly, a holonomic approximation of a family {su }, where u ranges over an interval I , is a family {αu }u∈I of closed curves such that αu is a holonomic ε-approximation of su for each u ∈ I . Our main result is the existence theorem for holonomic approximations. Theorem 1.1 Every curvature-like function admits a holonomic ε-approximation for every ε. The approximating curve S 1 → Rn+1 can be chosen so as to be an embedding. This result is not entirely trivial as it fails for families of curvature-like functions. Also, Remark 1.3 below shows that a curvature-like function, as defined here, may not admit a holonomic approximation by a curve in a prescribed Riemannian manifold. Proposition 1.2 There is a family {su } of c
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