Differential geometry of spatial curves for gauges
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Differential geometry of spatial curves for gauges Vitor Balestro1 · Horst Martini2 · Makoto Sakaki3
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract We derive Frenet-type results and invariants of spatial curves immersed in 3-dimensional generalized Minkowski spaces, i.e., in linear spaces which satisfy all axioms of finite dimensional real Banach spaces except for the symmetry axiom. Further on, we characterize cylindrical helices and rectifying curves in such spaces, and the computation of invariants is discussed, too. Finally, we study how translations of unit spheres influence invariants of spatial curves. Keywords Gauges · Generalized Minkowski spaces · Helix · Invariants · Spatial curves Mathematics Subject Classification 46B20 · 52A15 · 52A21 · 53A04 · 53A35
1 Introduction A function F ∶ ℝn → ℝ defined on the n-dimensional linear space ℝn is called a convex distance function, or a gauge, if it satisfies the following conditions:
F (x) ≥ 0 for x ∈ ℝn , and F(x) = 0 ⇔ x = 0, (1) F (𝜆x) = 𝜆F(x) for x ∈ ℝn , 𝜆 > 0, (2) Communicated by Yoshihiro Ohnita. * Makoto Sakaki sakaki@hirosaki‑u.ac.jp Vitor Balestro [email protected] Horst Martini [email protected]‑chemnitz.de 1
Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói 24210201, Brazil
2
Fakultät für Mathematik, Technische Universität Chemnitz, 09107 Chemnitz, Germany
3
Graduate School of Science and Technology, Hirosaki University, Hirosaki 036‑8561, Japan
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Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
(3) F (x + y) ≤ F(x) + F(y) for x, y ∈ ℝn. Having this notion, the space (ℝn , F) is called an n-dimensional generalized Minkowski space or gauge space (cf. [9]). It turns out that gauge spaces are characterized by the geometric property to have an arbitrary convex body B as unit ball, where the origin 0 is from the interior of B. (If 0 is even the center of symmetry of B, then we have the subcase of normed or Minkowski spaces, see [10, 11] for basic properties of these spaces, and [5] for results on curvatures of simple closed curves in such spaces.) Continuing the investigations from [1], which refer to curves in generalized Minkowski planes in the spirit of the normed subcase developed in [2], we will study now spatial curves in 3-dimensional gauge spaces. Our paper is organized as follows. In Sect. 2 we discuss Frenet-type equations and invariants for curves in 3-dimensional generalized Minkowski spaces. In Sect. 3 we characterize generalized cylindrical helices and rectifying curves in such spaces, and in Sect. 4 we investigate how to compute the invariants and give examples in a Randers space. In Sect. 5 we discuss how invariants of spatial curves are changing when the unit sphere changes its position with respect to 0 via parallel translation. It should be noticed that the paper [8] contains related (but mostly different) results; the differences and relations between the results in [8] and our results are always described in detail where t
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