Mannheim curves in the three-dimensional sphere

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1 Mannheim curves in the three-imensional sphere anju Kahraman, Mehmet Öner Manisa Celal Bayar University, Faculty of Arts an Sciences, Mathematics Department, Muraiye Campus, 5, Muraiye, Manisa, urkey. s: Abstract Mannheim curves are efine for immerse curves in -imensional sphere S. he efinition is given by consiering the geoesics of S. First, two special geoesics, calle principal normal geoesic an binormal geoesic, of S are efine by using Frenet vectors of a curve immerse in S. Later, the curve is calle a Mannheim curve if there exits another curve in S such that the principal normal geoesics of coincie with the binormal geoesics of. It is obtaine that if an form a Mannheim pair then there exist a constant λ an a nonconstant function µ such that λκ + µτ = where κ, τ are the curvatures of. Moreover, the relation between a Mannheim curve immerse in S an a generalize Mannheim curve in E is obtaine an a table containing comparison of Bertran an Mannheim curves in S is introuce. MSC: 5A Key wors: Spherical curves; generalize Mannheim curves; geoesics.. Introuction an Preliminaries Mannheim curves an Bertran curves are the most fascinating subject of the curve pairs efine by some relationships between two space curves. Mannheim curves are first efine by A. Mannheim in 878 [6]. In the Eucliean -space E, Mannheim curves are characterize as a kin of corresponing relation between two curves, such that the binormal vector fiels of coincie with the principal normal vector fiels of. hen an are calle Mannheim curve an Mannheim partner curve, respectively [9]. he main result for a curve to have a Mannheim partner curve in E is that there exists a constant λ such that τ κ = ( + λ τ ) hols, where κ, τ an s are the curvatures an arc length parameter of, s λ respectively [9]. Mannheim curves have been stuie by many mathematicians. Blum stuie a remarkable class of Mannheim curves []. In [7], Matsua an Yorozu have given a efinition of generalize Mannheim curve in Eucliean -space an introuce some characterizations an examples of generalize Mannheim curves. Later, Choi, Kang an Kim have efine Mannheim curves in - imensional Riemannian manifol []. Moreover, in the same paper, they have stuie Mannheim curves in -imensional space forms. Another type of associate curves is Bertran curves which first efine by French mathematician Saint-Venant in 85 by the property that two curves have common principal normal vector fiels at the corresponing points of curves [8]. Lucas an Ortega-Yagües have consiere Bertran curves in the three-imensional sphere S [5]. hey have given another S

2 efinition for space curves to be Bertran curves immerse in S an they have come to the result that a curve with curvatures κ, τ immerse in S is a Bertran curve if an only if either τ an is a curve in some unit two-imensional sphere constants λ, µ such that λκ + µτ = [5]. S () or there exit two In this stuy, we efine Mannheim curves in S by efining some special geoesics relate to the Frenet vectors of a curve immerse in S. We show that the angle between the tangent vector fiels of Mannheim curves is not constant while it is constant for Bertran curves. Moreover, we obtaine that a curve with curvatures κ, τ immerse in S is a Mannheim curve if there exist a constant λ an a non-constant function µ such that λκ + µτ =. We want to pointe out that this property hols for Bertran curves uner the conition that both λ an µ are constants.. Mannheim curves in the three-imensional sphere Before giving the main subject, first we give the following ata relate to the curves immerse in S. For this section, we refer the reaer to ref. [,5]. Let S ( r ) enote the three-imensional sphere in R of raius r, efine by S ( r) = ( x, x, x, x ) R xi = r, r >. i= Let = ( t) : I R S ( r) be an arc-length parameterize immerse curve in the -sphere S ( r ) an let {, N, B } an enotes the Frenet frame of an the Levi-Civita connection of S ( r ), respectively. hen, Frenet formulae of is given by = κ N N = κ + τ B B = τ N where κ an τ enote the curvature an torsion of, respectively. If Civita connection of R, then the Gauss formula gives X = X X,, r for any tangent vector fiel X χ( ). In particular, we have = κ N r N = κ + τ B B = τ N. A curve ( t) in stans for the Levi- S ( r ) is calle a plane curve if it lies in a totally geoesic two-imensional sphere S S which means that a curve ( t) in torsion τ is zero at all points [5]. S ( r ) is calle a plane curve if an only if its

3 Let ( t) an ( t) be two immerse curve in {, N, B } S ( r ) with Frenet frames {, N, B } an, respectively. A geoesic curve in S ( r ) starting at any point ( t) of an efine as u u t ( u) = cos ( t) + r sin B ( t), u R, r r is calle the binormal geoesic of an, similarly, a geoesic curve in S ( r ) starting at any point ( t) of an efine as u u t ( u) = cos ( t) + r sin N ( t), u R, r r is calle the principal normal geoesic of in S ( r ). Let be a regular smooth curve in Eucliean -space E efine by arc-length parameter s. he curve is calle a special Frenet curve if there exist three smooth functions k, k, k on e, e, e, e along the curve such that these satisfy the following an smooth frame fiel { } properties: i) he formulas of Frenet-Serret hols: e = e = k e e = k e + k e e = k e + k e e = k e where the prime (') enotes ifferentiation with respect to s. ii) he frame fiel {,,, } e e e e is orthonormal an has positive orientation. iii) he functions k an k are positive an the function k oesn t vanish. iv) he functions k, k an k are calle the first, the secon an the thir curvature e, e, e, e is calle the Frenet frame fiel functions of, respectively. he frame fiel { } on []. A special Frenet curve in E is a generalize Mannheim curve if there exists a special Frenet curve ˆ in E such that the first normal line at each point of is inclue in the plane generate by the secon normal line an thir normal line of ˆ at corresponing point uner a bijection φ from to ˆ. he curve ˆ is calle the generalize Mannheim mate curve of [7]. Now, we can introuce the main subject. First, we give the following efinition. Definition.. A curve in S ( r ) with non-zero curvature κ is sai to be a Mannheim curve if there exists another immerse curve = ( σ ) : J IR S ( r) an a one-to-one corresponence between an such that the principal normal geoesics of coincie with

4 the binormal geoesics of at corresponing points. We will say that is a Mannheim partner curve of ; the curves an are calle a pair of Mannheim curves. From this efinition it is clear that Mannheim partner curve can not a plane curve since the efinition given by consiering the binormal vector fiel of. For simplicity, we consier that the raius of the sphere S is an the curves taken on S are parameterize by the arc-length parameter. Let an ( σ ) be a pair of Mannheim curves. From Definition., we have a ifferentiable function a( s ) such that ( ) ( ) ( ) s( σ ) = cos a ( σ ) sin a B ( σ ) () where {, N, B } enotes the Frenet frame along ( ) σ, ( s( )) σ is the point in corresponing to ( σ ) an a( s ) is calle the angle function between the irection vectors an ( σ ). he function ( s ) is calle istance function in S an measures the istance between the points ( s( σ )) an ( σ ). Now, we can give the following proposition. Proposition.. Let an be a pair of Mannheim curves in S. In that case, we have the followings a) he angle function a( s ) is constant. b) he istance function ( s ) is constant. c) he angle θ between the tangent vector fiels at corresponing points is not constant. ) he angle between the binormal vectors fiels at corresponing points is constant. Proof. a) Since principal normal geoesic of an binormal geoesic of are common at corresponing points, we have s ( u) = B an s ( u) = N () u u u= u= a an then we obtain N = sin a ( σ ) + cos a B ( σ ) () ( ) ( ) where {, N, B } enotes the Frenet frame of. From (), the tangent vector to is given by ( s( σ )) = a sin ( a ) ( σ ) + cos ( a ) ( σ ) s () a cos a B ( σ ) + sin a τ N ( σ ) ( ) ( ) where s enotes ifferentiation with respect to s. Since, ( s( σ )) = s ( σ ) ( s( σ )) (5) σ we get = ( s( σ )), N = a ( sin ( a ) cos ( a ) ) (6) σ which gives that a =, i.e., a( s ) is constant.

5 b) Without loss of generality, for the angle function it can be taken as a π. hen the istance function ( s ) is given by = min{ a( s ), π a}, which is a constant function since a( s ) is constant. c) Let θ = θ ( σ ) enotes the angle between tangent vectors an, i.e., ( ) s( σ ), ( σ ) = cos θ ( σ ). Differentiating the left sie of this equality, it follows ( s ( σ )), ( σ ) = s ( σ ) κ N, +, κ N. (7) σ Moreover, from () an (5), we have ( s( σ )) = ( cos a + τ sin a N ) (8) s ( σ ) Writing (), () an (8) in (7), it follows κ τ sin a θ = s ( σ )sin θ ( σ ). (9) Since is not a plane curve from (9), it is clear that θ is not a constant. ) Using equality () an (), we can write B = sin a s( σ ) + cos a N ( σ ). () ( ) Since B, B =, the angle between binormal vectors is constant. σ heorem.. Let an be a pair of Mannheim curves in hol: tanθ a) τ = tan a τ sin a cosθ = cos a κ sin a sinθ b) ( ) c) ) θ a κ a a cos = cos sin cos sin θ = τ τ sin a S. hen the following equalities Proof. a) aking the covariant erivative in () an using (5), we obtain ( s( σ )) = cos θ s ( σ ) ( σ ) + sin θ s ( σ ) N ( σ ). () σ On the other han, by using Frenet equations we have ( s( σ )) = cos a ( σ ) + τ sin a N ( σ ) () σ where a = a is constant. he last two equations lea to s ( σ ) cosθ = cos a () s ( σ )sinθ = τ sin a () from which we conclue (a). b) We nee to write the Frenet frame of in terms of the Frenet frame of : ( ) ( ) ( σ ) = cos a s( σ ) + sin a N s( σ ) (5) 5

6 ( ) ( ) ( ) ( ) ( ) ( ) ( σ ) = cos θ s( σ ) + sin θ B s( σ ) (6) N ( σ ) = sin θ s( σ ) cos θ B s( σ ) (7) B ( σ ) = sin a s( σ ) + cos a N s( σ ). (8) From (a), it follows σ (s) cosθ = cos a κ sin a (9) σ (s)sinθ = τ sin a () which gives us (b). c) It is a consequence of Eqs. () an (9); cos θ = cos a κ sin a cos a. () ) Similarly, from Eqs. () an (), we have esire equality sin θ = τ τ sin a. () From (), one can consier that is a plane curve if an only if θ = or σ (s) =. Since σ is arc length parameter, it cannot be constant. Similarly, from Proposition.. (c), θ is not a constant. hen we can give the following corollary. Corollary.. A Mannheim curve cannot be a plane curve. heorem.. Relationship between arc-length parameters of curves an is given by s cos a cosθ τ sin a sinθ σ = +. () Proof. If equations (6) an (7) are written in (), we get equation (). heorem.. Let an be a pair of Mannheim curves in an a non-constant function µ such that S. hen there exists a constant λ = λκ + µτ. () Proof. Multiplying (9) an () by sinθ an cosθ, respectively, an equalizing obtaine results, gives that sinθ cos a = sinθ sin aκ + sin a cosθτ. (5) Writing λ = tan a an µ = tan a cotθ, from last equality we have = λκ + µτ. Corollary.. Both curvature κ an torsion τ of a Mannheim curve cannot be constant. Proof. Since is a Mannheim curve Eq. () hols. Let now assume that the curvature κ be a n constant. hen, from () we have τ = which is not a constant since µ is not a constant, µ where n = λκ is a real constant. Similarly, if it is assume that the torsion τ is constant, then by a similar way it is seen that κ is not constant. 6

7 heorem.. Let ( t) be a Mannheim curve in regular ifferentiable mapping s = s( t) with s ( t) S with constant curvature. hen there exists a > such that the curve ( ) t ( ( )) = t B s u u is an arc-length parametrize generalize Mannheim curve. Proof. By ifferentiating the curve ( t) three times an using Frenet formulae, we have κ = ε s τ, κ = s κ, κ = ε s, ε = ±. (6) (See [5]). From heorem., there exist constant λ an a non-constant µ such that = λκ + µτ hols. Let signs of λ an τ be same an consier a function s( t ) such that λτ s ( t) =, (7) τ + κ where λ is a non-zero real constant. Defining a constant by ε c =, λ from Eqs. ()-(6), we see that κ = c( κ + κ ) (8) hols for all s. hen from [7, heorem.], we have that ( t) is a generalize Mannheim curve. By consiering these characterizations an results obtaine for Bertran curves in [,5], we can give the following table giving the comparison of Bertran an Mannheim curves. t Characterizations Bertran Curves Mannheim Curves Angel between tangent vector fiels constant non-constant Angel between binormal vector fiels constant constant Main Characterization λκ + µτ = is a Bertran curve in S if an only if there exits two constant λ an µ such that λκ + µτ =. is a Bertran curve in S if there exit a constant λ an a non-constant function µ such that λκ + µτ = Curves, Curvatures an can be plane curves. an cannot be plane curves. Both κ an τ can be Both κ an τ cannot be constants. constants able. Comparison of Bertran an Mannheim curves in S. Some examples Example.. (ccr-curves) A C -special Frenet curve on S is sai to be a ccr-curve on S if κ its intrinsic curvature ratio is a constant number []. Let now etermine special ccr-curve τ on S which is also Mannheim curve. First assume that is a ccr-curve with non-constant 7

8 curvature an non-constant torsion. hen, we have κ = cτ for a non-zero constant c. Writing this equality in () gives us c κ =, τ =. tan a( c + cot θ ) tan a( c + cot θ ) hen we conclue that the ccr-curve on curve. Example.. (Conical helix) A twiste curve in S given by curvatures as given above is a Mannheim S with non-constant curvatures is sai to be conical helix if both the curvature raius κ an the torsion raius τ evolve linearly along the curve [5]. hen the curvature an torsion of curve are given by δ κ =, τ = s + r s + r, respectively, where r, r, an δ are constants. aking = δ, we see that () hols for a constant λ = δ an a non-constant function Mannheim curve. µ = s + r + r s + r r r ( ) ( ) s + r, i.e., is a Example.. (General helix) A twiste curve in S is a general helix if there exists a constant b such that τ = bκ ± []. Let now etermine general helices in S which are also µ Mannheim curves. Writing the conition τ = bκ ± in (), it follows κ =. hen we λ + bµ µ b( µ ) have that a general helix in S with curvatures κ =, τ = ± is a Mannheim λ + bµ λ + bµ curve. Example.. (Curve with constant curvatures) Consier C curve on S () given by the parametrization = cos s, sin s, cos s, sin s for all s R. he curvatures are compute as κ = an τ = ε, ε = ± []. hen from Corollary., is not a Mannheim curve. References [] Barros, M., General helices an a theorem of Lancret, Proceeings of the American Mathematical Society 5 (997) [] Blum, R., A Remarkable class of Mannheim-curves, Cana. Math. Bull., 9(966), -8. [] Choi, J., Kang,. an Kim, Y., Mannheim Curves in -Dimensional Space Forms, Bull. Korean Math. Soc. 5() ()

9 [] Kim, C.Y., Park, J.H., Yorozu, S., Curves on the unit -sphere S () in the Eucliean -space R, Bull. Korean Math. Soc., 5(5) () [5] Lucas, P., Ortega-Yagües, J., Bertran Curves in the three-imensional sphere, Journal of Geometry an Physics, 6 () 9 9. [6] Mannheim, A., Paris C.R. 86 (878) [7] Matsua, H., Yorozu, S., On Generalize Mannheim Curves in Eucliean -space, Nihonkai Math. J., (9) 56. [8] Saint-Venant, J.C., Mémoire sur les lignes courbes non planes, Journal Ecole Polytechnique (85) 76. [9] Wang, F., Liu, H., Mannheim partner curves in -Eucliean space, Mathematics in Practice an heory, 7() (7) -. [] Wong, Y.C., Lai, H.F., A critical examination of the theory of curves in three imensional ifferential geometry, ohoku Math. J. 9 (967). 9