Solving of Curvatures of the Frenet Frame by Normal, Geodesic and Geodesic Torsion Equations


Bayrak Torbalı Ü. G. , İyigün E.

8th INTERNATIONAL CONFERENCE on RECENT ADVANCES in PURE AND APPLIED MATHEMATICS (ICRAPAM 2021), Muğla, Turkey, 24 - 27 September 2021, pp.149-150

  • Publication Type: Conference Paper / Summary Text
  • City: Muğla
  • Country: Turkey
  • Page Numbers: pp.149-150

Abstract

Minkowski space is one of the best mathematical demonstration to explain Einstein’s relativity theory. Minkowksi have thought that the elements in three-dimensional space can be classified into three categories as space-like, time-like and light-like [1-4]. < 𝑥, 𝑦 >= −𝑥1𝑦1 + 𝑥2𝑦2 + 𝑥3𝑦3 , the negativity (the absence of a positively defined inner product) has provided flexibility for using the space [5].Initially, it can be shown that the normal curvature, geodesic curvature and geodesic torsion as 𝑘𝑛, 𝑘𝑔 and τg respectively [6]. In E3 , T is tangent vector field of an α unitspeed curve, 𝑘𝑔 = ‖𝐷𝑇𝑇‖ = ‖ 𝑑 2𝛼 𝑑𝑡2‖. Here, this equation states the geodesic curvature corresponding to α(s) pointfor α curve in E3 [7]. 𝜗𝑝 ∈ 𝑇p(M), 𝑘𝑛 = 𝑘𝑝 (𝜗𝑝 ) =< 𝑆( 𝜗𝑝 ‖𝜗𝑝‖ ), 𝜗𝑝 ‖𝜗𝑝‖ >, 𝑘𝑝 (𝜗𝑝 ) identified by the equivalence is the normal curvature of the surface M in the 𝜗𝑝 direction at the point p [7]. Here, S is a shape operator.In this study, normal curvature, geodesic curvature and geodesic torsion have been defined in Euclidean space for Frenet frame. Possible solutions of normal, geodesic and geodesic torsion equations on Frenet frame have been investigated.