On Natural Lift of a Curve
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1 Pure Mathematical Sciences, Vol. 1, 2012, no. 2, On Natural Lift of a Curve Evren ERGÜN Ondokuz Mayıs University, Faculty of Arts and Sciences Department of Mathematics, Samsun, Turkey eergun@omu.edu.tr Mustafa ÇALIŞKAN Gazi University, Faculty of Sciences Department of Mathematics, Ankara, Turkey mustafacaliskan@gazi.edu.tr Abstract In this study, the Frenet vector fiel T,N,B,curvature κ and torsion τ of the natural lift α of a curve α are calculated in terms of those of α in R 3. The same study has been done in R 3 1. Mathematics Subject Classification: 51B20, 53A15, 53A04 Keywor: Natural Lift, Frenet Frame, Curvature, Torsion 1 Introduction and Preliminary Notes Let α : I R 3 be a parametrized curve. We denote by {T (s),n(s),b(s)} the moving Frenet frame along the curve α, where T,N and B are the tangent, the principal normal and the binormal vector of the curve α, respectively. Let α be a reguler curve in R 3.Then If α is a unit speed curve,then T = α α,n= B T, B = α α α α, [6]. T = α,n= α,b= T N, [6]. α
2 82 E. ERGÜN and M. ÇALIŞKAN Let α be a unit speed space curve with curvature κ and torsion τ and let Frenet vector fiel of α be {T,N,B}. Then, Frenet formulas are given by T = κn, N = κt + τb, B = τn, [1]. where κ = T,N and τ = N,B. For any unit speed curev α : I R 3, we call W (s) =τκb (s) the Darboux vector field of α, [1]. Let M be a hypersurface in R 3 and let α : I M be a parametrized curve. α is called an integral curve of X if d (α (s)) = X (α (s)) (for all s I), [1]. where X is a smooth tangent vector field on M. We have TM = P M T P M = χ (M), where T P M is the tangent space of M at P and χ (M) is the space of vector fiel on M. For any parametrized curve α : I M, α : I TM given by α (s) = α (s),α (s) = α (s) α(s), [5]. is called the natural lift of α on TM.Thus, we can write dα = d ) (α (s) α(s) = D α (s) α (s) where D is the Levi-Civita connection on R 3. Let α : I R 3 1 be a parametrized curve. We denote by {T (s),n(s),b(s)} the moving Frenet frame along the curve α, where T,N and B are the tangent, the principal normal and the binormal vector of the curve α, respectively. Let α be a unit speed timelike space curve with curvature κ and torsion τ. Let Frenet vector fiel of α be {T,N,B}. In this trihedron, T is timelike vector field, N and B are spacelike vector fiel. Then, Frenet formulas are given by [4] T = κn N = κt + τb B = τn. Let α be a unit speed spacelike space curve with a spacelike binormal. In this trihedron, we assume that T and B are spacelike vector fiel and N is a timelike vector field.then, Frenet formulas are given by [4] T = κn N = κt + τb B = τn. Let α be a unit speed spacelike space curve with a timelike binormal. In this trihedron, we assume that T and N are spacelike vector fiel and B is a timelike vector field. Then, Frenet formulas are given by [4]
3 On natural lift of a curve 83 T = κn N = κt + τb B = τn. Let M be a hypersurface in R 3 1 and let α : I M be a parametrized curve. α is called an integral curve of X if d (α (s)) = X (α (s)) (for all s I) [2] where X is a smooth tangent vector field on M. We have TM = T P M = χ (M) P M where T P M is the tangent space of M at P and χ (M) is the space of vector fiel on M. For any parametrized curve α : I M, α : I TM given by α (s) = α (s),α (s) = α (s) α(s).[3] is called the natural lift of α on TM.Thus, we can write dα = d ) (α (s) α(s) = D α (s) α (s) where D is the Levi-Civita connection on R ON NATURAL LIFT OF A CURVE For any parametrized curve in R 3 α : I M, α : I TM α (s) = α (s),α (s) = α (s) α(s) given by is called the natural lift of α on TM. We denote by { T (s), N (s), B (s) } the moving Frenet frame along the curve α, where T,N and B are the tangent, the principal normal and the binormal vector of the curveα, respectively. Corollary 1 Let α be the natural lift of α in R 3 and be a reguler curve. Then.
4 84 E. ERGÜN and M. ÇALIŞKAN Let α be a space curve with curvature κ and torsion τ. Then κ = τ = N, B. T, N and Corollary 2 Let α be the natural lift of α with curvature κ and torsion τ.then κ (s) = κ2 (s)+τ 2 (s) For any parametrized curve in R 3 1 α : I M, α : I TM given by α (s) = α (s),α (s) = α (s) α(s) is called the natural lift of α on TM. We denote by { T (s), N (s), B (s) } the moving Frenet frame along the curve α, where T,N and B are the tangent, the principal normal and the binormal vector of the curve α, respectively. Corollary 3 Let α be a unit speed timelike space curve and α be the natural lift of α.then B (s) = τ (s) κ (s). Corollary 4 Let α be a unit speed timelike space curve and the natural lift α of the curve α be a space curve with curvature κ and torsion τthen κ (s) = κ2 (s) τ 2 (s) Corollary 5 Let α be a unit speed spacelike space curve with a spacelike binormal and α be the natural lift of α.then N (s) = κ (s) τ (s).
5 On natural lift of a curve 85 Corollary 6 Let α be a unit speed spacelike space curve with a spacelike binormal and the natural lift α of the curve α be a space curve with curvature κ and torsion τ. Then κ (s) = κ2 (s)+τ 2 (s) (s) τ (s) κ (s) τ (s), τ (s) = κ W Corollary 7 Let α be a unit speed spacelike space curve with a timelike binormal and α be the natural lift of α.then. Corollary 8 Let α be a unit speed spacelike space curve with a timelike binormal and the natural lift α of the curve α be a space curve with curvature κ and torsion τ.then References κ (s) = κ2 (s)+τ 2 (s) [1] B. O Neill, Elementery Differential Geometry Academic Press, New York and London, [2] B. O Neill. Semi-Riemannian Geometry, with applications to relativity.academic Press, New York, (1983). [3] E, Ergün., Çalışkan, M., On Geodesic Sprays In Minkowski 3-Space, International Journal of Contemp. Math. Sciences, Vol. 6, no.39, (2011), [4] J. Walrave, Curves and Surfaces in Minkowski Space K. U. Leuven Faculteit Der Wetenschappen, [5] M. Çalışkan, A.İ. Sivridağ, and H.H. Hacısalihoğlu, Some Characterizations for the natural lift curves and the geodesicspray, Communications,Fac. Sci. Univ. Ankara Ser. A Math. 33 (1984), Num. 28, [6] M. P. Do Carmo, Differential Geometry of Curve Surfaces. Prentice-Hall, Inc., Englewood Cliffs, New Jersey Received: November, 2011
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