Başlık: Biharmonic curves in contact geometryYazar(lar):KOCAYIĞIT, Hüseyin ; HACISALIHOĞ LU, H. HilmiCilt: 61 Sayı: 2 Sayfa: 035-043 DOI: 10.1501/Commua1_0000000678 Yayın Tarihi: 2012 PDF

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IS S N 1 3 0 3 –5 9 9 1

BIHARMONIC CURVES IN CONTACT GEOMETRY

HÜSEYIN KOCAYI µGIT AND H. HILMI HACISALIHO µGLU

Abstract. We study biharmonic curves in contact geometry whose mean cur-vature vector …eld is in the kernel of Laplacian. We give some results for bi-harmonic curves in Sasakian 3-space. We also give some characterizations for Legendre curves in the same space.

1. Introduction and Preliminaries.

Let M be a smooth manifold. A contact form on M is a 1-form such that (d )n _^ _{6= 0. A manifold M together with a contact form is called a contact}

manifold [4,10]. The distribution D de…ned by the Pha¢ an equation = 0 is called the contact structure determined by . That is,

D = f x 2 (M)j : (M ) ! C1(M; R); _{(X) = 0g}

(see, for instance, [4,10]). The maximum dimension of integral submanifold of D is (dim M 1)=2. An integral submanifold of D of maximum dimension is called a Legendre submanifold of (M; ) [3].

The reel vector …eld (killing vector …eld) is de…ned by ( ) = 1; d ( ; :) = 0 (see [10]).

On a contact manifold (M; ), there exist an endomorphism …eld and a Rie-mannian metric g satisfying

2₌ _{I +} _;

g( X; Y ) = g(X; Y ) (X) (Y ); d (X; Y ) = 2g(X; Y );

for all vector …elds X and Y on M . The structure tensors ( ; ; g) is called the associated almost contact structure of [10].

A contact manifold (M; ; ; ; g) is said to be a Sasaki manifold if M satis…es 2000 Mathematics Subject Classi…cation. 53A04.

Key words and phrases. Biharmonic curve, geodesic, circular helix, heneral helix, contact manifold..

c 2 0 1 2 A n ka ra U n ive rsity

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(rX )Y = g(X; Y ) (Y )X

(see [10]).

Since is a globally de…ned unit vector …eld, the contact manifold M admits a Lorentz metric. In fact let us de…ne h by

h = g 2 :

Then h is a Lorentz metric which satis…es the following formulae: h(X; ) = (X);

h( X; Y ) = h(X; Y ) + (X) (Y ); d (X; Y ) = 2h(X; Y ):

We call the Lorentz metric h by associated Lorentz metric of M (see [8], [9]). Let us de…ne by r the Levi-Civita connection of the Lorentz metric h. Then r is related to the Levi-Civita connection rg of g by the following formula:

rXY = rgXY + 2 ( (X) Y + (Y ) X) :

Now let (M; ; ; ; g) be a Sasaki manifold. Then the associated Lorentz metric h satis…es the following equation (Theorem 3 in [2]):

(rX )Y = h(X; Y ) + (Y )X;

rX = X:

The reel vector …eld is globally de…ned timelike killing vector …eld on the Lorentz manifold (M; h). The resulting manifold (M; ; ; ; g) is called a Lorentz-Sasaki manifold or Lorentz-Sasakian spacetime (see [2], [6]).

Now let M3_{= (M; ; ;} _{; g) be a contact 3-manifold with an associated metric}

g. A curve _{= (s) : I ! M parameterized by the arclength parameter is said to} be a Legendre curve if is tangent to contact distribution D of M . It is obvious that is Legendre if and only if ( 0_{) = 0.}

Let be a Legendre curve on M3_{. Then we can take a Frenet frame fV}1; V2; V3g

so that V1= 0 and V3= .

Now we assume that M is a Sasaki manifold. Then the following equality is de…ned

(rX )Y = g(X; Y ) (Y )X; X; Y 2 (M):

The Frenet-Serret formulae of are given explicitly by 2 4 r 0_V 1 r0V2 r0V3 3 5 = 2 4 0 0 01 0 1 0 3 5 2 4 VV12 V3 3 5 : (1.1)

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The function is called the curvature of . Namely, every Legendre curve has constant torsion 1 (see [1]). In particular, a curve parametrized by the arclength is said to be a geodecis if = 0. Note that if = 0, then automatically = 0.

More generally a curve with constant curvature and zero torsion is called a (Rie-mannian) circle. Geodesics are regarded as Riemannian circles of zero curvatures.

A circular helix is a curve whose curvature and torsion are constants. Geodesics and Riemannian circles are regarded as degenerate helices. Helices which are neither geodesics nor circles are frequently called proper helices.

Let us denote by the Laplace-Beltrami operator of and by H the mean curvature vector …eld along .

The Frenet-Serret formulae of imply that the mean curvature vector …eld H is given by

H = r0 0= r0V1= V2; (1.2)

where is the curvature of .

The Laplace-Beltrami operator of is de…ned by

H = _r02H (1.3)

(see [5], [7]).

De…nition 1.1. A unit speed Legendre curve _{= I ! M}3 _{on Sasakian}

3-manifold is said to be biharmonic if H = 0(i.e., H = 2 _{= 0).}

Chen and Ishikawa [5] classi…ed biharmonic curves in semi-Euclidean space En v.

In this paper we shall give the characterizations of biharmonic curves in contact geometry in terms of curvature.

2. Biharmonic Curves on Sasakian 3-Manifolds with Rimennian Metric. In this section we give the characterizations for biharmonic curves on Sasakian 3-manifolds. By using the obtained results we give some conditions for these curves to be helix.

Theorem 2.1. Let be a unit speed Legendre curve on Sasakian 3-manifold and let be a real constant. Then Legendre curve is a circular helix if and only if the following di¤ erential equation satis…es

H + H = 0 (2.1)

where = 2 _1.

Proof. By the use of (1.1), (1.2) and (1.3) we get that

H = 3 0V1+ ( 3+ 00)V2 2 0V3; (2.2)

H = V2: (2.3)

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Theorem 2.2. Let be a unit speed Legendre curve on Sasakian 3-manifold. Then satis…es the following di¤ erential equation

r03V1+ 1r02V1+ 2r0V1+ 3V1= 0; (2.4) where 1= 2 0 ; 2= 2+ 1 002 _{2 (} 02 3 ; 3= 0:

Proof. By the use of Frenet-Serret formulae (1.1), we have

r02V1= 2V1+ 0V2+ V3: (2.5)

By di¤erentiating (2.5) with respect to arc parameter we obtain

r03V1= 2 0V1 2r0V1+ 00V2+ 0r0V2+ 0V3+ r0V3: (2.6)

From the second equation of (1.1) we get

V3= r0V2+ V1; (2.7)

and from the …rst equation of (1.1) we have V2=

1

r0V1: (2.8)

Substituting (2.8) into (2.7) we obtain V3= 1 0 r0V1+ 1 r02V1+ V1: (2.9) Di¤erentiating (2.9) gives r0V3= 1 r03V1+ 2 1 0 r02V1+ " + 1 00# r0V1+ 0V1: (2.10)

Substituting (2.8) into the third equation of (1.1) we have r0V3=

1

r0V1: (2.11)

Substituting (2.11) into the third equation of (2.10) and doing the regulations we have

r03V1+ 1r02V1+ 2r0V1+ 3V1= 0;

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1= 2 0

; 2= 2+ 1

002 _{2 (} 02

3 ; 3= 0; (2.12)

that …nishes the proof.

Corollary 1. Let be a unit speed Legendre curve on Sasakian 3-manifold. If the Legendre curve is geodesic, then the curve is a circular helix.

Proof. Let _{be a geodesic curve. Then r}0V1= 0 which gives that is constant

i.e., is a circular helix.

Corollary 2. Let be a unit speed Legendre curve on Sasakian 3-manifold. If the Legendre curve is a circular helix, then the di¤ erential equation characterizing the curve is

r03V1+ ( 2+ 1)r0V1= 0: (2.13)

Proof. The proof follows from (2.4) immediately.

r03V2+ 1r02V2+ 2r0V2+ 3V2= 0; (2.14) where 1= 00 0; 2= (1 + 2_); 3= 3 0 002₎ 0 :

Proof. The proof is obtained immediately by considering the similar way used in the proof of Theorem 2.1.

r03V2+ ( 2+ 1)r0V2= 0:

Proof. The proof follows from (2.4) immediately.

r03V3+ 1r02V3+ 2r0V3+ 3V3= 0; (2.15)

where

1=

0

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Proof. From the third equation of equation (1.1) we have

r02V3= r0V2: (2.16)

Substituting the second equation of equation (1.1) into (2.16) we get

r02V3= V1 V3: (2.17)

Di¤erentiating (2.17) gives

r03V3= 0V1+ r0V1 r0V3: (2.18)

From the third equation of Equation (1.1) we have

r0V3= V2: (2.19)

From the …rst equation of Equation (1.1) and (2.19) we get

r0V1= r0V3: (2.20)

Similarly, from the second equation of Equation (1.1) we have V1=

1 V3

1

r0V2: (2.21)

From (2.16) and (2.21) we obtain V1=

1 V3

1

r02V3: (2.22)

Then, writing (2.20) and (2.22) in the (2.18) we have (2.15).

r03V3+ (1 + 2)r0V3= 0: (2.23)

Proof. The proof is clear from Theorem 2.4.

3. Biharmonic Curves on Sasakian 3-Manifolds with Lorentzian Metric. Let be a Legendre curve on Sasakian 3-manifold M . Then according to the Lorentzian metric the Frenet-Serret formulae of are given explicitly by

2 4 r 0_V 1 r0V2 r0V3 3 5 = 2 4 0 0 0" 0 1 0 3 5 2 4 VV12 V3 3 5 ; (3.1)

where " = 1 is the torsion of [3]. The Laplacian operator and the mean curvature H of are de…ned, respectively, by

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= _"r02; (3.2) and

H = r0V1: (3.3)

Theorem 3.1. Let be a unit speed Legendre curve on Sasakian 3-manifold and let be a real constant. Then Legendre curve is a circular helix if and only if the following di¤ erential equation satis…es

H = H; then is a constant and

= " + 2: (3.4)

Proof. By (3.1), (3.2) and (3.3), we get (3.4). The converse statement of Theorem 3.1. is also …ne.

Corollary 5. Let be a unit speed Legendre curve on Sasakian 3-manifold (M; g). Then is a Legendre circular helix whose Killing vector …eld is timelike (respectively spacelike) if and only if H = H where = 1 2_{(respectively} _{= 1 +} 2_).

Proof. The proof is easily seen by Theorem 3.1.

Corollary 6. Let be a unit speed Legendre curve on Sasakian 3-manifold (M; g). Then is a Legendre biharmonic circular helix, whose Killing vector …eld is timelike (respectively spacelike) if and only if = 1 (respectively, = i) .

Proof. The proof is easily seen by Theorem 5.

Theorem 3.2. Let be a unit speed curve with Lorentzian metric on Sasakian 3-manifold (M; g). Then satis…es the following di¤ erential equation

r03V1+ 1r02V1+ 2r0V1+ 3V1= 0; (3.5) where 1= 2 0 ; 2= 2+ " 002 _{2 (} 02 3 ; 3= 0:

Proof. We get (3.5) by using (3.1).

Corollary 7. Let be a unit speed Legendre curve on Sasakian 3-manifold with Lorentzian metric. If the Legendre curve is geodesic then the equation character-izing the curve is

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Proof. Let be a geodesic curve. Then the proof is clear from Corollary 1. Corollary 8. Let be a unit speed Legendre curve on Sasakian 3-manifold. If the Legendre curve is a circular helix then the di¤ erential equation characterizing the curve is

r03V1+ ( 2+ ")r0V1= 0: (3.7)

Proof. By (3.5), we get (3.7).

r03V2+ 1r02V2+ 2r0V2+ 3V2= 0; (3.8) where 1= 00 0; 2= 2_{+ ";} 3= 3 0 00_{(1 +} 2₎ 0 :

Proof. We get (3.8) by using (3.1).

r03V2+ ( 2+ ")r0V2= 0: (3.9)

Proof. By (3.8), we get (3.9).

r03V3+ 1r02V3+ 2r0V3+ 3V3= 0; (3.10) where 1= 0 ; 2= 2+ "; 3= " 0 : Proof. We get (3.10) by using (3.1).

r03V3+ ( 2+ ")r0V3= 0: (3.11)

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Özet:Bu çal¬¸smada, contact geometride ortalama eµgrilik vektör alan¬, Laplasyan operatörünün çekirdeµginde olan biharmonik eµgrileri inceleriz. Sasakian 3-uzay¬nda biharmonic eµgriler için baz¬sonuçlar veririz. Bununla birlikte, ayn¬ uzayda Legendre eµgriler için baz¬ karakterizasyonlar¬veririz.

References

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[10] K. Yano, M. Kon, Structures on manifolds, Series in pure mathematics. Volume 3, (1984). Current address : Hüseyin Kocayiµgit, Department of Mathematics, Faculty of Art&Sci. Celal Bayar University, 45047 Manisa, TURKEY, H. Hilmi Hac¬salihoµglu, Ankara University, Faculty of Science, Department of Mathematics, Tando¼gan, Ankara, TURKEY

E-mail address : huseyin.kocayigit@cbu.edu.tr, hacisali@science.ankara.edu.tr URL: http://communications.science.ankara.edu.tr/index.php?series=A1