tailieunhanh - P-elastica in the 3-dimensional lorentzian space forms

R Huang worked the p-elastic in a Riemannian manifold with constant sectional curvature. In this work, we solve the Euler-Lagrange equation by quadrature and study the Frenet equation of the p-elastica by using the Killing field in the three dimensional Lorentzian space forms. | Turk J Math 30 (2006) , 33 – 41. ¨ ITAK ˙ c TUB p-Elastica in the 3-Dimensional Lorentzian Space Forms Nevin G¨ urb¨ uz Abstract R Huang worked the p-elastic in a Riemannian manifold with constant sectional curvature [1]. In this work, we solve the Euler-Lagrange equation by quadrature and study the Frenet equation of the p-elastica by using the Killing field in the three dimensional Lorentzian space forms. 1. Introduction Definition Let L be a 3-dimensional Lorentzian space. If (x1 , x2, x3 ) and (y1 , y2 , y3 ) are the components of X and Y with respect to an allowable coordinate system, then hX, Y i |L = x1 y1 + x2 y2 − x3 y3 which is called a Lorentzian inner product. Furthermore, a Lorentz exterior product X×Y is given by X × Y = (x2 y3 − x3 y2 , x3 y1 − x1 y3 , x2y1 − x1 y2 ). Then, for any x ∈ L3 it holds ([2]) 2 hX × Y, X × Y i = hX, Y i − hX, Xi hY, Y i . Mathematics Subject Classification: 53A04, 35Q51, 53B99, 53A35. 33 ¨ ¨ GURB UZ Definition A semi-Riemannian manifold M has constant curvature if its sectional curvature function is constant. If M constant curvature C , then ([3]) Rxy z = C{hz, xi y − hz, yi x}. −−→ − → Definition The norm of X ∈ R31 is denoted by kXk and defined as ([3]) −−→ kXk = r D → − →E − X , X . Theorem Let γ(s) be a unit speed curve in R31 , s the arclength parameter. Consider the Frenet frame {T = γ 0 , N, B} attached to the curve γ = γ(s) such that is T is the unit tangent vector field, N is the principal normal vector field and B = T × N is the binormal vector field. The Frenet -Serret formulas is given by ∇T T = T 0 0 ∇T N = N 0 = −ε1 κ ∇T B = B 0 0 ε2 κ 0 ε2 τ 0 T −ε3 τ N , 0 B () where hT, T i = ε1 , hN, N i = ε2 , hB, Bi = ε3 . ∇ is the semi Riemannian connection on M and κ = κ(s) and τ = τ (s) are the curvature and the torsion functions of γ, respectively. 2. Killing Fields This section is taken from [4], [5]. Let γ(t) be a nonnull .