In this paper, we introduce special Smarandache curves and obtain Frenet apparatus of a Smarandache curve in three dimensional Lie groups with a bi-invariant metric

(1)

C om mun. Fac. Sci. U niv. A nk. Ser. A 1 M ath. Stat.

Volum e 68, N umb er 1, Pages 1175–1185 (2019) D O I: 10.31801/cfsuasm as.000000

ISSN 1303–5991 E-ISSN 2618-6470

http://com munications.science.ankara.edu.tr/index.php?series= A 1

SMARANDACHE CURVES IN THREE DIMENSIONAL LIE GROUPS

O. ZEK·I OKUYUCU, CANER DE ¼G·IRMEN, AND Ö. GÖKMEN YILDIZ

Abstract. In this paper, we introduce special Smarandache curves and obtain Frenet apparatus of a Smarandache curve in three dimensional Lie groups with a bi-invariant metric. Moreover, we give some relations between a helix or a slant helix curve and its Smarandache curve in three dimensional Lie Groups.

1. Introduction

In the classical di¤erential geometry, curves theory is a most important work area. Special curves and their characterizations have been studied for a long time and are still being studied. The application of special curves is seen in nature, mechanic tools, computer aided design and computer graphics etc.

One of the special curves is Smarandache curve, whose position vector is com- posed by Frenet frame vectors on an other regular curve. In [1], Ahmad introduced some special Smarandache curves in the Euclidean space. Then, in [2], Smaran- dache curves were examined according to Darboux frame in Euclidean 3-space.

Also, some researchers studied Smarandache curves in Minkowski space, [8, 10].

Furthermore, in [9], Ta¸sköprü and Tosun have investigated Smarandache curves according to Sabban Frame in Euclidean 3-space.

The degenerate semi-Riemannian geometry of Lie group has been studied by Çöken and Çiftçi [6]. In this work, they obtained a naturally reductive homoge- neous semi-Riemannian space using the Lie group. Then, Çiftçi [5] de…ned general helices in three dimensional Lie groups with a bi-invariant metric and obtained a generalization of Lancret’s theorem. Also, a relation between the geodesics of the so-called cylinders and general helices is given in the same study.

In [11], Okuyucu et al. de…ned slant helices in a three dimensional Lie group G with a bi-invariant metric as a curve : I R !G whose normal vector …eld makes a constant angle with a left invariant vector …eld. Also, they de…ned Bertrand curves in [12]. Gök et al., in [7], studied Mannheim curves in three dimensional Lie

Received by the editors: May 25, 2018; Accepted: September 28, 2018.

2010 Mathematics Subject Classi…cation. Primary 53A04; Secondary 22E15.

Key words and phrases. Smarandache curves, Lie groups, helices.

c 2 0 1 9 A n ka ra U n ive rsity C o m m u n ic a tio n s Fa c u lty o f S c ie n c e s U n ive rs ity o f A n ka ra -S e rie s A 1 M a t h e m a t ic s a n d S ta t is t ic s

1175

(2)

groups. Bozkurt et al., in [3], investigated the characterizations of the rectifying, normal and osculating curves in a three dimensional compact Lie group with a bi- invariant metric. As a physical application, Okuyucu et al., [13], obtained Spinor Frenet equations of curves and Körp¬nar, [14, 15], investigated a new version of the energy of curves in three dimensional compact Lie groups.

In this paper, we introduce special Smarandache curves in three dimensional Lie groups with a bi-invariant metric and obtain Frenet apparatus of a Smarandache curve in three dimensional Lie groups.

2. Preliminaries

By G we shall denote a Lie group with a bi-invariant metric h ; i in three dimensional Euclidean space. If g is the Lie algebra of G; then we know that g is isomorphic to TeG where e is neutral element of G: Let r be the Levi-Civita connection of Lie group G: If h ; i is a bi-invariant metric on G, we have

hX; [Y; Z]i = h[X; Y ] ; Zi (1)

and

r^XY = 1 2[X; Y ] for all X; Y and Z 2 g:

Let : I R !G be an arc-lenghted regular curve and fV1; V2;V3g be an orthonormal basis ofg: In this case, any two vector …elds and along the curve can be given as =P3

i=1 iVi and =P3

i=1 iVi where _i: I ! R and ⁱ: I ! R are smooth functions. Furthermore, the Lie bracket of two vector …elds and is given

[ ; ] = X3 i;j=1

i j[Vi; Vj]

and the covariant derivative of along the curve with the notation r ^p is given as follows

r ^p = _ +1

2[T; ] (2)

where T = ⁰ and _ =P3 i=1

d _i

dtV_i: Note that if is a left-invariant vector …eld to the curve then = 0 (see [4] for details).

Let fT; N; B; ; g denote the Frenet apparatus of the curve , then we have

= T in G where T, N and B are called, the tangent, the principal normal,_ the binormal vector …elds, respectively. And also, is the curvature and is the torsion of :

De…nition 1. Let : I R !G be a parametrized curve with (T; N; B; ; ) then

G= 1

2h[T; N] ; Bi (3)

(3)

or

G = 1

2 ²

DhT; T• i

; _T E

+ 1

4 ² h

T; _T i 2

(see [5]).

Proposition 2. Let : I R !G be an arc length parametrized curve with fT; N; Bg. Then the following equalities hold,

[T; N] = h[T; N] ; Bi B = 2 ^GB;

[T; B] = h[T; B] ; Ni N = 2 ^GN;

(see [11]).

Theorem 3. Let : I R !G be an arc length parametrized curve with fT; N; B; ; g.

Then the harmonic curvature function of is de…ned by

h = ^G

where G =¹₂h[T; N] ; Bi (see [11]).

Theorem 4. Let : I R ! G be a parametrized curve with (T; N; B; ; ). Then is a general helix if and only if

= c + G

where c is a constant (see [5]). Hence a curve is a general helix if and only if its harmonic curvature function h is a constant function.

Remark 5. Let be a helix with Frenet apparatus (T; N; B; h) in three dimensional Lie group. Then the axis of the curve can be given as

X = h

p1 + h²T + 1

p1 + h²B (4)

or considering the constant harmonic curvature function h = cot , = constant, can be given

X = cos T + sin B (5)

with the help of reference [5].

Theorem 6. Let : I R !G be an arc length parametrized curve with fT; N; B; ; g.

Then is a slant helix if and only if

N = 1 + h²

3 2

h^p = tan '

is a constant, where h is harmonic curvature function of the curve and ' 6= ₂ is a constant (see [11]).

(4)

3. Smarandache Curves in a three dimensional Lie group

In this section, we de…ne Smarandache curves and obtain Frenet apparatus of these curves in three dimensional Lie groups with a bi-invariant metric.

Unless otherwise stated, throughout the paper : I R !G is an arc-lenghted regular curve with the Frenet apparatus fT; N; B; ; g in three dimensional Lie group G with a bi-invariant metric:

De…nition 7. Let be a curve in G: TN Smarandache curve can be de…ned as (s ) = 1

p2(T(s) + N(s)) : (6)

Now, we compute Frenet invariants of TN Smarandache curve. Di¤erentiating Eq. (6) with respect to s, we get

p= d ds

ds ds = 1

p2[ _T(s) + _N(s)]

and

T ds ds =p

2( T(s) + N(s) + hB(s)) ; where

ds ds = p

2

p2 + h²: (7)

And so, the tangent vector of can be written as follow, T (s ) = T(s) + N(s) + hB(s)

p2 + h² : (8)

By di¤erentiating Eq. (8), we have dT

ds ds

ds = A T(s) + B N(s) + C B(s) (2 + h²)³²

(9) where

A = 2 + h² + hh^p;

B = 1 + h² 2 + h² hh^p; C = h + h^p 2 + h² h²h^p: Substituting (7) in (9), we get

T_ =

p2

(2 + h²)²(A T(s) + B N(s) + C B(s)) :

Then, the curvature and principal normal vector …eld of curve are respectively,

= T_ = p2 (2 + h²)²

q

A² + B² + C²

(5)

and

N (s ) = 1

q

A² + B² + C² (A T(s) + B N(s) + C B(s)) : So, the binormal vector of curve is

B (s ) = T N

= 1

qf(C hB ) T(s) + (C + hA ) N(s) (A + B ) B(s)g where =p

2 + h²and q =q

A² + B² + C²:

In order to calculate the torsion of the curve , we di¤erentiate the ^p

q= 1

p2 ( ² ^p)T(s) + ( ^p ²(1 + h²))N(s) + ( ²h + ^ph + h^p)B(s) and thus

ppp=l T(s) + m N(s) + n B(s) p2

where

l = 3 ^p(1 h) + 2 ²h^p;

m = ³(1 h)(1 + h²) + ^q(1 h) 2 ^ph^p h^q; n = 3 ^ph(1 h) + ²h^p(1 3h):

Thus we compute

= det ^p; ^q; ^ppp

p q 2 ;

=

p2 h ²+ h^p l + h^pm + ²n ( h ²+ h^p)²+ ( h^p)²+ ^{4 4} :

Corollary 8. Let be a curve and be the TN Smarandache curve of in G:

If the curve is a helix with the axis X, then the tangent vector of the curve is perpendicular the vector …eld X.

Proof. It is obvious using the equations (4) and (8).

De…nition 9. Let be a curve in G: TB Smarandache curve can be de…ned as

!(s!) = 1

p2(T(s) + B(s)) : (10)

Now, we compute Frenet invariants of TB Smarandache curve. Di¤erentiating Eq. (10) with respect to s, we get

!^p= d!

ds_! ds!

ds = 1

p2[ _T(s) + _B(s)] (11)

(6)

and

T!

ds!

ds = p

2(1 h) N(s) where

ds_! ds = p

2(1 h) : (12)

And so, the tangent vector of ! can be written as follow,

T!(s!) = N(s): (13)

By di¤erentiating Eq. (13), we have dT!

ds!

ds_!

ds = T(s) + hB(s): (14)

Substituting (12) in (14), we get T_!=

p2

(1 h)( T(s) + hB(s)) :

Then, the curvature and principal normal vector …eld of curve ! are respectively,

!= T_! = p2 (1 h)

p1 + h²

and

N!(s_!) = 1

p1 + h²T(s) + h

p1 + h²B(s):

So, the binormal vector of curve ! is B!= T! N!= h

p1 + h²T(s) + 1

p1 + h²B(s):

In order to calculate the torsion of the curve !, we di¤erentiate Eq. (11)

!^q= 1 p2

2(1 h)T(s) + ^p(1 h) h^p N(s) + ²h(1 h)B(s) and thus

!^ppp= l!T(s) + m!N(s) + n!B(s) p2

where

l!= 3 ^p(1 h) + 2 ²h^p;

m!= ³(1 h)(1 + h²) + ^q(1 h) 2 ^ph^p h^q; n!= 3 ^ph(1 h) + ²h^p(1 3h):

(7)

Thus we compute

!= det (!^p; !^q; !^ppp) k!^p !^qk² ;

!=

p2(n_!+ hl_!)

3(1 h)²(1 + h²):

Corollary 10. Let be a curve and ! be the TB Smarandache curve of in G. If the curve is a slant helix with the axis X, then the curve ! is a helix.

Furthermore, the axis of the curve ! is the same axis of . Proof. One can see by considering the Eq. (13).

De…nition 11. Let be a curve in G: NB Smarandache curve can be de…ned as (s ) = 1

p2(N(s) + B(s)) : (15)

Now, we compute Frenet invariants of NB Smarandache curve. Di¤erentiating Eq. (15) with respect to s, we get

p= d ds

ds ds = 1

p2[ _N(s) + _B(s)]

and

T ds ds = p

2[ T(s) hN(s) + hB(s)];

where

ds ds =p

2

p1 + 2h²: (16)

And so, the tangent vector of can be written as follow, T = T(s) hN(s) + hB(s)

p1+2h² : (17)

ds ds

ds =A T(s) + B N(s) + C B(s) (1 + 2h²)³²

(18) where

A = h 1 + 2h² + 2hh^p; B = 1 + h² 1 + 2h² h^p;

C = h² 1 + 2h² + h^p: Substituting (16) in (18), we get

T =_

p2

(1 + 2h²)²(A T(s) + B N(s) + C B(s)) :

(8)

Then, the curvature and principal normal vector …eld of curve are respectively,

= T_ =

p2 (1 + 2h²)²

q

A²+ B²+ C² and

N = 1

q

A²+ B²+ C²(A T(s) + B N(s) + C B(s)) : So, the binormal vector of curve is

B (s ) = T (s ) N (s )

= 1

q[ h (C + B ) T(s) + (C + hA ) N(s) + ( B + hA ) B(s)]

where =p

1 + 2h² and q = q

A²+ B²+ C²:

In order to calculate the torsion of the curve , we di¤erentiate the ^p

q= 1

p2f( ^p+ ²h)T(s) + ² 1 + h² ^ph h^p N(s) + ²h²+ ^ph + h^p B(s)g

and thus

ppp=l T(s) + m N(s) + n B(s) p2

where

l = ^q+ 3 ^ph + 2 ²h^p+ ³(1 + h²);

m = ³h(1 + h²) 3 ^p ^qh 2 ^ph^p h^q 3 h ( h)^p; n = ³h(1 + h²) 3 h ( h)^p+ ^qh + 2 ^ph^p+ h^q: Thus we compute

= det ^p; ^q; ^ppp

p q 2 ;

=

p2 h ²l + h^pm + ²+ h^p n 2 ²h^p(h^p+ ²) + ^{4 4}(1 + h²) :

Corollary 12. Let be a curve and be the NB Smarandache curve of in G.

Proof. This is an immediate consequence of the equations (4) and (17).

(9)

De…nition 13. Let be a curve in G: TNB Smarandache curve can be de…ned as (s ) = 1

p3(T(s) + N(s) + B(s)) : (19) Now, we compute Frenet invariants of TNB Smarandache curve. Di¤erentiating Eq. (19) with respect to s, we get

p= d ds

ds ds = 1

p3

T(s) + _N(s) + _B(s)_ and

T ds ds = p

3[ T(s) + (1 h) N(s) + hB(s)]

where,

ds ds =

p2 p3

p1 h + h²: (20)

And so, the tangent vector of can be written as follow, T (s ) = T(s) + (1 h) N(s) + hB(s)

p2p

1 h + h² : (21)

ds ds

ds = A T(s) + B N(s) + C B(s) 2p

2 (1 h + h²)³²

(22) where

A = 2 1 h + h² (1 h) h^p(1 2h) ;

B = 2 1 h + h² h^p h² + h^p(1 2h) (1 h) ; C = 2 1 h + h² h (1 h) + h^p + hh^p(1 2h) : Substituting (20) in (22), we get

T =_

p3

4 (1 h + h²)² A T(s) + B N(s) + C B(s) :

Then, the principal curvature and principal normal vector …eld of curve are respectively,

= T_ =

p3 4 (1 h + h²)²

q

A²+ B²+ C² and

N = 1

qA²+ B²+ C² A T(s) + B N(s) + C B(s) :

(10)

So, the binormal vector of curve is

B = T N

= 1

p2 q (1 h) C hB T(s) + C + hA N(s) B + (1 h) A B(s) where =p

1 h + h²and =q

A²+ B²+ C²:

In order to calculate the torsion of the curve , we di¤erentiate

q= 1

p3f ²(1 h) ^p T(s) + ² 1 + h² + ^p(1 h) h^p N(s) + ²h(1 h) + ^ph + h^p B(s)g

and thus

ppp=l T(s) + m N(s) + n B(s)p 3

where

l = ^q 3 ^p(1 h) + 2 ²h^p+ ³ 1 + h² ;

m = ³(1 h) 1 + h² 3 ^p+ ^q(1 h) 2 ^ph^p h^q 3 h ( h)^p; n = ³h 1 + h² 2 ²hh^p+ h (1 h) 3 ^p+ + ^qh + 2 ^ph^p+ h^q: Thus we compute

= det ^p; ^q; ^ppp

p q 2 ;

=

p3 2 ²h ²+ h^p l + h^pm + 2 ^{2 2}+ h^p n 4 ^{4 2}( h^{4 2}+ h^p(1 + h)) + 3 ³(h^p)² :

Corollary 14. Let be a curve and be the TNB Smarandache curve of in G.

Proof. It is obvious using the equations (4) and (21).

Acknowledgement. We would like to thank the reviewers for their valuable com- ments to improve the work.

References

[1] Ahmad, T. Ali, Special Smarandache Curves in the Euclidean Space, International J.Math.

Combin., (2010), 2, 30-36.

[2] Bekta¸s, Ö. and Yüce, S., Special Smarandache Curves According to Darboux Frame in E³, Romanian J. of Math and Comp. Sci., (2013), 3(1), 48-59.

[3] Bozkurt, Z., Gök, ·I., Okuyucu, O. Z. and Ekmekci, F. N., Characterizations of rectifying, normal and osculating curves in three dimensional compact Lie groups, Life Science Journal, (2013), 10(3), 819-823.

(11)

[4] Crouch, P. and Silva Leite, F., The dynamic interpolation problem:on Riemannian manifoldsi Lie groups and symmetric spaces, J. Dyn. Control Syst., (1995), 1(2), 177-202.

[5] Çiftçi, Ü., A generalization of Lancert’s theorem, J. Geom. Phys., (2009), 59, 1597-1603.

[6] Çöken, A. C., and Çiftci Ü., A note on the geometry of Lie groups, Nonlinear Anal. TMA, (2008), 68, 2013-2016.

[7] Gök, ·I., Okuyucu, O. Z., Ekmekci, N. and Yayl¬, Y., On Mannheim Partner Curves in three Dimensional Lie Groups, Miskolc Mathematical Notes, (2014), 15(2), 467-479.

[8] Gürses, B. N., Bekta¸s, Ö. and Yüce, S., Special Smarandache Curves in R³1, Commun. Fac.

Sci. Univ. Ank. S~er. A1 Math. Stat., (2016), 65(2), 143-160.

[9] Ta¸sköprü, K. and Tosun, M., Smarandache Curves on S²,Boletim da Sociedade Paranaense de Matemática 3 srie., (2014), 32(1), 51-59.

[10] Turgut, M. and Y¬lmaz, S., Smarandache Curves in Minkowski Space-time, International J.

Math. Combin., (2008), 3, 51-55.

[11] Okuyucu, O. Z., Gök, ·I., Yayl¬, Y. and Ekmekci, N., Slant Helices in three Dimensional Lie Groups, Appl. Math.Comput., (2013), 221, 672-683.

[12] Okuyucu, O. Z., Gök, ·I., Yayl¬, Y. and Ekmekci, N., Bertrand Curves in three Dimensional Lie Groups, Miskolc Mathematical Notes, (2017), 17(2), 999-1010.

[13] Okuyucu, Z., Y¬ld¬z, Ö. G. and Tosun, M., Spinor Frenet Equations in Three Dimensional Lie Groups, Adv. Appl. Cli¤ ord Algebras, (2016), 26(4), 1341-1348.

[14] Körp¬nar, T., A New Version of Energy for Slant Helix with Bending Energy in the Lie Groups, Journal of Science and Arts, (2017), 4(41), 721-730.

[15] Körp¬nar, T., A New Version of the Energy of Tangent Indicatrix with Dynamics System in Lie Group, Di¤ er. Equ. Dyn. Syst., (2018), https://doi.org/10.1007/s12591-018-0413-y.

Current address : O. Zeki Okuyucu: Bilecik ¸Seyh Edebali University, Faculty of Sciences and Arts, Department of Mathematics, Bilecik, Turkey.

E-mail address : osman.okuyucu@bilecik.edu.tr

ORCID Address: http://orcid.org/0000-0003-4379-0252

Current address : Caner De¼girmen: Bilecik ¸Seyh Edebali University, Faculty of Sciences and Arts, Department of Mathematics, Bilecik, Turkey.

E-mail address : caner.deo@gmail.com

Current address : Ö. Gökmen Y¬ld¬z: Bilecik ¸Seyh Edebali University, Faculty of Sciences and Arts, Department of Mathematics, Bilecik, Turkey.

E-mail address : ogokmen.yildiz@bilecik.edu.tr

ORCID Address: https://orcid.org/0000-0002-2760-1223