Frenet Curve Couples in Three Dimensional Lie Groups

Osman Zeki Okuyucu

Department of Mathematics,Faculty of Science and Arts,Bilecik¸Seyh Edebali University,Bilecik,Turkey

ABSTRACT In this study, we examine the possible relations between the Frenet planes of any given two curves in three dimensional Lie groups with leたinvariant metrics.We explain these possible relations in nine cases and then introduce the conditions that must be met to coincide with the planes of these curves in nine theorems.

KEYWORDS Curves;lie groups;curvatures;Frenet plane

1 Introduction

The theory of curves has an important role in differential geometry studies.In the theory of curves,one of the interesting problems is to investigate the relations between two curves.The Frenet elements of the curves have an effective role in the solution of the problem.

For example,if the principal normal vectors coincide at the corresponding points of the curvesαandβ, the curve couple {α,β} is called the Bertrand curve couple in a three-dimensional Euclidean space[1,2].Similarly,if tangent vectors coincide,the curve couple{α,β}is called the involute-evolute curve couple.Also,if the normal vector of the curveαcoincides with the bi-normal vector of the curveβ,the curve couple{α,β}is called the Mannheim curve couple[3].

In a three-dimensional Lie groupGwith a bi-invariant metric,Çiftçi has defined general helices[4].Also Okuyucu et al.have obtained slant helices and Bertrand curves inG[5,6].Gök et al.have investigated Mannheim curves inG[7].Recently, Yampolsky et al.have examined helices in three dimensional Lie groupG,with left-invariant metric[8].Also,many applications of curves theory are studied and still have been investigated in three dimensional Lie groups(see[9-12],etc.).

Karaku¸s et al.have examined the possibility of whether any Frenet plane of a given space curve in a three-dimensional Euclidean space is also any Frenet plane of another space curve in the same space[13].

In this study,we examine the possible relations between the Frenet planes of given two curves in three dimensional Lie groups with left invariant metrics.

2 Preliminaries

LetGbe a three dimensional Lie group with left-invariant metric 〈,〉 and let g denote the Lie algebra ofGwhich consists of the all smooth vector fields ofGinvariant under left translation.There are two classes of three dimensional Lie groups:

1.If the group is unimodular,we have a(positively oriented)orthonormal frame of left-invariant vector fields{e1,e2,e3},such that the brackets satisfy

whereλiare called structure constants.The constants

are called connection coefficients.

2.If the group is nonunimodular,we have an othonormal frame{e1,e2,e3},such that

see[14].

Using the Koszul formula the covariant derivatives ∇eiejcan be found as in the following tables:

for unimodular and nonunimodular cases,respectively.

The cross-products of the vectorse1,e2,e3are defined by the following equalities:

in three dimensional case.For unimodular and nonunimodular groups, we have ∇eiek=μ(ei)×ek,and so

for any vector fieldX,whereμis a affine transformation as follows:

for unimodular and nonunimodular cases,respectively(see[8]).

We have ∇eiek=μ(ei)×ekfor both groups,and so

for any vector fieldX.

Letγbe an arc-lengthed curve on the group andT=be the unit tangent vector field.For any vector fieldξ°γ,using the Eq.(1),we get

LetT,NandBbe the vectors of the standard Frenet frame of the curveγ.We get the following equations with the help of Eq.(2):

Along the curveγ,we can define a new frame{τ,v,β},which is called dot-Frenet Frame,by

wherek0=||.By definition0=||.

Proposition 2.1.The dot-Frenet frame{τ,v,β}satisfies dot-Frenet formulas,namely

The Frenet and the dot-Frenet frames are connected by

τ=T,v=cosαN+sinαB,β=-sinαN+cosαB,

whereα=α(s)is the angle function(see[8]).

Proposition 2.2.The transformationμ(T)can be given by

with respect to the Frenet frame{T,N,B}.

Define a group-curvaturekGand a group-torsionof a curve by

respectively.As a consequence of(4),the dot-curvature and the dot-torsion of a curve can be expressed in terms of the group-curvaturekG,group-torsionof a curve,and angle functionαby

(see[8]).

3 Frenet Curve Couples in Three Dimensional Lie Groups

Letζ:I⊆R →Gandη:⊆R →Gbe curves with arc-length parametersandrespectively,in three dimensional Lie groupGwith left-invariant metric.We denote Frenet apparatus of the curvesζandηwith{T,N,B,k0,0,α}and,respectively.We know thatSp{T,N},Sp{N,B}andSp{T,B} are the osculating plane, the normal plane and the rectifying plane of the curveζ,respectively.And similarly,are the osculating plane, the normal plane and the rectifying plane of the curveη,respectively.Now we investigate the possible cases and the relations between the Frenet planes of any given two curves in three dimensional Lie groups with left invariant metrics in a step by step manner:

Case 1:We assume that,osculating plane of the curveζis the osculating plane of the curveη,that isAs in Fig.1,this relationship exists at the corresponding points of along the curvesζandη.

Figure 1:Osculating planes of the curves ζ and η

So we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(5)with the help of Eqs.(3)and(4),we get

We know thatBis parallel to,sinceIf we multiply the Eq.(6)withB,we get

And so,we have

By using Eq.(7),we can set

whereψis smooth angle function betweenTandonIand

By using the Eqs.(9)and(10),we obtain

and

Calculating the dot-derivative of the Eq.(8)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(11),withand,respectively,we get

By using Eqs.(12)and(13),we obtain

Thus we introduce the following theorem:

Theorem 3.1.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The osculating planes of these curves coincide if and only if there exist real valued non-zero functionsaandbonI,such that

whereψis the angle betweenTandat the corresponding points ofζandη.

Case 2:We assume that,osculating plane of the curveζis the normal plane of the curveη,that isSp{T,N}=Sp{,}.As in Fig.2,this relationship exists at the corresponding points of along the curvesζandη.

Thus,we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(14)with the help of Eqs.(3)and(4),we get

Figure 2:Osculating plane of the curve ζ and normal plane of the curve η

We know thatBis parallel to, sinceIf we multiply the Eq.(15)withT,NandB,respectively,we get

If we multiply the Eq.(16)withand,respectively,we get

whereψis the smooth angle function betweenTand.By using the Eqs.(17)and(18),we obtain

that is

Thus we introduce the following theorem:

Theorem 3.2.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The osculating plane of the curveζand the normal plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that:

whereψis the angle betweenTandat the corresponding points ofζandη.

Case 3:We assume that,osculating plane of the curveζis the rectifying plane of the curveη,that isAs in Fig.3,this relationship exists at the corresponding points of along the curvesζandη.

Figure 3:Osculating plane of the curve ζ and rectifying plane of the curve η

Thus,we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(19)with the help of Eqs.(3)and(4),we get

We know thatBis parallel to, sinceIf we multiply the Eq.(20)withB,we get

And so,we have

By Eq.(21),we can set

whereψis smooth angle function betweenTandonIand

By using the Eqs.(23)and(24),we obtain

and

Calculating the dot-derivative of the Eq.(22)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(25),withand,respectively,we get

By using Eqs.(26)and(27),we obtain

Thus we introduce the following theorem:

Theorem 3.3.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The osculating plane of the curveζand the rectifying plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that

whereψis the angle betweenTandat the corresponding points ofζandη.

Case 4:We assume that,normal plane of the curveζis the osculating plane of the curveη,that isAs in Fig.4,this relationship exists at the corresponding points of along the curvesζandη.

Figure 4:Normal plane of the curve ζ and osculating plane of the curve η

So we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(28)with the help of Eqs.(3)and(4),we get

We know thatTis parallel to, sinceIf we multiply the Eq.(29)withT,we get

And so,we have

By the Eq.(30),we can set

whereψis smooth angle function betweenandNonIand

By using Eqs.(32)and(33),we obtain

Calculating the dot-derivative of the Eq.(31)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(34),withand,respectively,we get

By using Eqs.(35)and(36),we obtain

Thus we introduce the following theorem:

Theorem 3.4.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The normal plane of the curveζand the osculating plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that

whereψis the angle betweenandNat the corresponding points ofζandη.

Case 5:We assume that,normal plane of the curveζis the normal plane of the curveη,that isAs in Fig.5, this relationship exists at the corresponding points of along the curvesζandη.

Figure 5:Normal planes of the curves ζ and η

Thus,we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(37)with the help of Eqs.(3)and(4),we get

We know thatTis parallel tosinceIf we multiply the Eq.(38)withT,NandB,respectively,we getak0cosα+bk0sinα=1-r,

And so,we have the equation=T.If we calculate the dot-derivative of this equation with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(39)withandrespectively,we get

whereψis the smooth angle function betweenNand.By using the Eqs.(40)and(41),we obtain

This means that,

Thus we introduce the following theorem:

Theorem 3.5.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The normal planes of these curves coincide if and only if there exist real valued non-zero functionsaandbonI,such that:

whereψis the angle betweenNandat the corresponding points ofζandη.

Case 6:We assume that, normal plane of the curveζis the rectifying plane of the curveη, that isAs in Fig.6,this relationship exists at the corresponding points of along the curvesζandη.

Figure 6:Normal plane of the curve ζ and rectifying plane of the curve η

Thus,we have following relations between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(42)with the help of Eqs.(3)and(4),we get

We know thatTis parallel to, sinceIf we multiply the Eq.(43)withT,we get

And so,we have

By the Eq.(44),we can set

whereψis smooth angle function betweenandNonIand

By using Eqs.(46)and(47),we obtain

and

Calculating the dot-derivative of the Eq.(45)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(48),withand,respectively,we get

By using Eqs.(49)and(50),we obtain

Thus we introduce the following theorem:

Theorem 3.6.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The normal plane of the curveζand the rectifying plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that:

whereψis the angle betweenandNat the corresponding points ofζandη.

Case 7:We assume that,rectifying plane of the curveζis the osculating plane of the curveη,that isAs in Fig.7,this relationship exists at the corresponding points of along the curvesζandη.

Figure 7:Rectifying plane of the curve ζ and osculating plane of the curve η

So we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(51)with the help of Eqs.(3)and(4),we get

We know thatNis parallel tosinceIf we multiply the Eq.(52)withN,we get

And so,we have

By the Eq.(53),we can set

whereψis smooth function betweenTandonIand

By using Eqs.(55)and(56),we obtain

and

Calculating the dot-derivative of the Eq.(54)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(57)withandrespectively,we get

By using Eqs.(58)and(59),we obtain

Thus we introduce the following theorem:

Theorem 3.7.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The rectifying plane of the curveζand the osculating plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that:

whereψis the angle betweenTandat the corresponding points ofζandη.

Case 8:We assume that, rectifying plane of the curveζis the normal plane of the curveη, that isAs in Fig.8,this relationship exists at the corresponding points of along the curvesζandη.

Figure 8:Rectifying plane of the curve ζ and normal plane of the curve η

Thus,we have following relation between the curvesζandη:

whereaandbare real valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(60)with the help of Eqs.(3)and(4),we get

We know thatNis parallel tosinceIf we multiply the Eq.(61)withT,NandB,respectively,we get

If we multiply the Eq.(62)withand,respectively,we get

whereψis the smooth angle function between theTandBy using Eqs.(63)and(64),we obtain

that is

Thus we introduce the following theorem:

Theorem 3.8.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The rectifying plane of the curveζand the normal plane of the curveηcoincide if and only if there exist real valued non-zero functionsaandbonI,such that

whereψis the angle betweenTandat the corresponding points ofζandη.

Case 9:We assume that,rectifying plane of the curveζis the rectifying plane of the curveη,that isAs in Fig.9,this relationship exists at the corresponding points of along the curvesζandη.

Figure 9:Rectifying planes of the curves ζ and η

Thus,we have following relation between the curvesζandη:

whereaandbare real-valued non-zero functions ofs.

Calculating the dot-derivative of the Eq.(65)with the help of Eqs.(3)and(4),we get

We know thatNis parallel to, sinceIf we multiply the Eq.(66)withN,we get

And so,we have

By the Eq.(67),we can set

whereψis smooth angle function betweenTandonIand

By using Eqs.(69)and(70),we obtain

and

Calculating the dot-derivative of the Eq.(68)with the help of Eqs.(3)and(4),we get

If we multiply the Eq.(71)withand,respectively,we get

By using Eqs.(72)and(73),we obtain

Thus we introduce the following theorem:

Theorem 3.9.Letζ:I⊆R →Gandη:⊆R →Gbe two arc-length parametrized curves with the Frenet apparatus{T,N,B,k0,0,α}andrespectively,in three dimensional Lie groupGwith left-invariant metric.The rectifying planes of these curves coincide if and only if there exist real valued non-zero functionsaandbonI,such that

whereψis the angle betweenTandat the corresponding points ofζandη.

4 Conclusions

It is well known that every smooth curves have a moving Frenet frame.This paper examines the relations between Frenet planes of two smooth curves in three dimensional Lie groups with leftinvariant metric.There are nine possible relations that can occur.For each cases,we give conditions by nine theorems as above.These results are generalizations for relations between Frenet planes of two curves in three dimensional Euclidean spaces.By the paper’s results,one will be able to investigate of special curve couples in three-dimensional Lie groups with left-invariant metric and correlate their results.

Funding Statement:The author received no specific funding for this study.

Conflicts of Interest:The author declares that he has no conflicts of interest to report regarding the present study.