We study curves of AW(k)-type in the Lie group G with a bi-invariant metric. Also, we characterize general helices in terms of AW(k)-type curve in the Lie group G.

1. Introduction

The geometry of curves and surfaces in a 3-dimensional Euclidean space ℝ3 represented for many years a popular topic in the field of classical differential geometry. One of the important problems of the curve theory is that of Bertrand-Lancret-de Saint Venant saying that a curve in ℝ3 is of constant slop; namely, its tangent makes a constant angle with a fixed direction if and only if the ratio of torsion τ and curvature κ is a constant. These curves are said to be general helices. If both τ and κ are nonzero constants, the curve is called cylindrical helix. Helix is one of the most fascinating curves in science and nature. Scientists have long held a fascinating, sometimes bordering on mystical obsession for helical structures in nature. Helices arise in nanosprings, carbon nanotubes, α-helices, DNA double and collagen triple helix, the double helix shape is commonly associated with DNA, since the double helix is structure of DNA.

The problem of Bertrand-Lancret-de Saint Venant was generalized for curves in other 3-dimensional manifolds—in particular space forms or Sasakian manifolds. Such a curve has the property that its tangent makes a constant angle with a parallel vector field on the manifold or with a Killing vector field, respectively. For example, a curve α(s) in a 3-dimensional space form is called a general helix if there exists a Killing vector field V(s) with constant length along α and such that the angle between V and α′ is a non-zero constant (see [1]). A general helix defined by a parallel vector field was studied in [2]. Moreover, in [3] it is shown that general helices in a 3-dimensional space form are extremal curvatures of a functional involving a linear combination of the curvature, the torsion, and a constant. General helices also called the Lancret curves are used in many applications (e.g., [4–7]).

The notion of AW(k)-type submanifolds was introduced by Arslan and West in [8]. In particular, many works related to curves of AW(k)-type have been done by several authors. For example, in [9, 10] the authors gave curvature conditions and charaterizations related to these curves in ℝn. Also, in [11] they investigated curves of AW(k) type in a 3-dimensional null cone and gave curvature conditions of these kinds of curves. However, to the author’s knowledge, there is no article dedicated to studying the notion of AW(k)-type curves immersed in Lie group.

In this paper, we investigate curvature conditions of curves of AW(k)-type in the Lie group G with a bi-invariant metric. Moreover, we characterize general helices of AW(k)-type in the Lie group G.

2. Preliminaries

Let G be a Lie group with a bi-invariant metric 〈,〉 and D the Levi-Civita connection of the Lie group G. If 𝔤 denotes the Lie algebra of G, then we know that 𝔤 is isomorphic to TeG, where e is identity of G. If 〈,〉 is a bi-invariant metric on G, then we have
(2.1)〈X,[Y,Z]〉=〈[X,Y],Z〉,DXY=12[X,Y]
for all X, Y, Z∈𝔤.

Let α:I⊂ℝ→G be a unit speed curve with parameter s and {V1,V2,…,Vn} an orthonrmal basis of 𝔤. In this case, we write that any vector fields W and Z along the curve α as W=∑i=1nwiVi and Z=∑i=1nziVi, where wi:I→ℝ and zi:I→ℝ are smooth functions. Furthermore, the Lie bracket of two vector fields W and Z is given by
(2.2)[W,Z]=∑i=1nwizj[Vi,Vj].
Let Dα′W be the covariant derivative of W along the curve α, V1=α′, and W′=∑i=1nwi′Vi, where wi′=dwi/ds. Then we have
(2.3)Dα′W=W′+12[V1,W].

A curve α is called a Frenet curve of osculating order d if its derivatives α′(s), α′′(s), α′′′(s),…,α(d)(s) are linearly dependent and α′(s),α′′(s),α′′′(s),…,α(d+1)(s) are no longer linearly independent for all s. To each Frenet curve of order d one can associate an orthonormal d-frame V1(s),V2(s),V3(s),…,Vd(s) along α (such that α′(s)=V1(s)) called the Frenet frame and the functions k1,k2,…,kd-1:I→ℝ said to be the Frenet curvatures, such that the Frenet formulas are defined in the usual way:
(2.4)DV1V1(s)=k1(s)V2(s),DV1V2(s)=-k1(s)V1(s)+k2(s)V3(s),⋮DV1Vi(s)=-ki-1(s)Vi-1(s)+ki(s)Vi+1(s),DV1Vi+1(s)=-ki(s)Vi(s).
If α:I→G is a Frenet curve of osculating order 3 in G, then we define
(2.5)k-2(s)=12〈[V1,V2],V3〉.

Proposition 2.1.

Let α be a Frenet curve of osculating order 3 in G. Then one has
(2.6)[V1,V2]=〈[V1,V2],V3〉V3=2k-2V3,[V1,V3]=〈[V1,V3],V2〉V2=-2k-2V2,[V2,V3]=〈[V2,V3],V1〉V1=2k-2V1.

Proof.

Let α be a Frenet curve of osculating order 3 with the Frenet frame {V1,V2,V3}. Since [V1,V2]=a1V1+a2V2+a3V3, taking the inner product with V1, V2, and V3, respectively, we have a1=a2=0 and 〈[V1,V2],V3〉=a3. Thus, we find
(2.7)[V1,V2]=〈[V1,V2],V3〉V3.
From (2.5), we get
(2.8)[V1,V2]=2k-2V3.
By using the above similar method, we can obtain [V1,V3]=-2k-2V2 and [V2,V3]=2k-2V1.

Remark 2.2.

Let G be a 3-dimensional Lie group with a bi-invariant metric. Then it is one of the Lie groups SO(3), S3 or a commutative group, and the following statements hold (see [6, 12]).

If G is SO(3), then k-2(s)=1/2.

If G is S3≅SU(2), then k-2(s)=1.

If G is a commutative group, then k-2(s)=0.

Proposition 2.3.

Let α be a Frenet curve of osculating order 3 in G. Then one has
(2.9)α′(s)=V1(s),α′′(s)=k1(s)V2(s),α′′′(s)=-k12(s)V1(s)+k1′(s)V2(s)+k1(s)τ1(s)V3(s),α′′′′(s)=-3k1(s)k1′(s)V1(s)+[k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)]V2(s)+(2k1′(s)τ1(s)+k1(s)τ1′(s))V3(s),
where τ1(s)=k2(s)-k-2(s).

Proof.

Let α be a Frenet curve of osculating order 3 in G. Then we have
(2.10)α′′(s)=d2αds2=V1′(s)=DV1V1(s)-12[V1(s),V1(s)]=k1(s)V2(s).
This implies that
(2.11)α′′′(s)=k1′(s)V2(s)+k1(s)V2′(s)=k1′(s)V2(s)+k1(s)(DV1V2(s)-12[V1(s),V2(s)])=k1′(s)V2(s)+k1(s)(-k1(s)V1(s)+k3(s)-k-2(s)V3(s))=-k12(s)V1(s)+k1′(s)V2(s)+k1(s)(k2(s)-k-2(s))V3(s).
Also, we have the following:
(2.12)α′′′′(s)=-2k1(s)k1′(s)V1(s)+k1′′(s)V2(s)+(k1(s)k2(s)-k1(s)k-2(s))′V3(s)-k12(s)V1′(s)+k1′(s)V2′(s)+k1(s)(k2(s)-k-2(s))V3′(s)=-2k1(s)k1′(s)V1(s)+k1′′(s)V2(s)+(k1(s)k2(s)-k1(s)k-2(s))′V3(s)-k12(s)(DV1V1(s)-12[V1(s),V1(s)])+k1′(s)(DV1V2(s)-12[V1(s),V2(s)])+k1(s)(k2(s)-k-2(s))(DV1V3(s)-12[V1(s),V3(s)])=-3k1(s)k1′(s)V1(s)+[k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)]V2(s)+(2k1′(s)τ1(s)+k1(s)τ1′(s))V3(s).

Notation. Let we put
(2.13)N1(s)=k(s)V2(s),N2(s)=k1′(s)V2(s)+k1(s)τ1(s)V3(s),N3(s)=[k1′′(s)-k13-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)]V2(s)+(2k1′(s)τ1(s)+k1(s)τ1′(s))V3(s).

3. Curves of AW(k)-Type

In this section, we consider the properties of curves of AW(k)-type in the Lie group G.

of type weak AW(2) if they satisfy
(3.1)N3(s)=〈N3(s),N2*(s)〉N2*(s),

of type weak AW(3) if they satisfy
(3.2)N3(s)=〈N3(s),N1*(s)〉N1*(s),
where
(3.3)N1*(s)=N1(s)∥N1(s)∥,N2*(s)=N2(s)-〈N2(s),N1*(s)〉N1*(s)∥〈N2(s),N1*(s)〉N1*(s)∥.

Definition 3.2 (see [<xref ref-type="bibr" rid="B3">8</xref>]).

The Frenet curves of osculating order 3 are

of type AW(1) if they satisfy N3(s)=0,

of type AW(2) if they satisfy
(3.4)∥N2(s)∥2N3(s)=〈N3(s),N2(s)〉N2(s),

of type AW(3) if they satisfy
(3.5)∥N1(s)∥2N3(s)=〈N3(s),N1(s)〉N1(s).

From the definitions of type AW(k), we can obtain the following propositions.

Proposition 3.3.

Let α be a Frenet curve of osculating order 3. Then α is of weak AW(2)-type if and only if
(3.6)k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)=0.

Proposition 3.4.

Let α be a Frenet curve of osculating order 3. Then α is of weak AW(3)-type if and only if
(3.7)2k1′(s)τ1(s)+k1(s)τ1′(s)=0.

Proposition 3.5.

Let α be a Frenet curve of osculating order 3. Then α is of AW(1)-type if and only if
(3.8)k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)=0,k12(s)τ1(s)=c,
where c is a constant.

Proposition 3.6.

Let α be a Frenet curve of osculating order 3. Then α is of type AW(2) if and only if
(3.9)k1′(s)(2k1′(s)τ1(s)+k1(s)τ1′(s))=k1(s)τ1(s)(k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s))=0.

Proposition 3.7.

Let α be a Frenet curve of osculating order 3. Then α is of type AW(3) if and only if
(3.10)k12(s)τ1(s)=c,
where c is a constant.

4. General Helices of AW(k)-Type

In this section, we study general helices of AW(k)-type in the Lie group G with a bi-invariant metric and characterize these curves.

Definition 4.1 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

Let α:I→G be a parameterized curve. Then α is called a general helix if it makes a constant angle with a left-invariant vector field.

Note that in the definition the left-invariant vector field may be assumed to be with unit length, and if the curve α is parametrized by arc-lengths, then we have
(4.1)〈α′(s),X〉=cosθ,
for X∈𝔤, where θ is a constant.

If G is a commutative group ℝ3, then Definition 4.1 reduces to the classical definition (see [14]). Since a left-invariant vector field in G is a Killing vector field, Definition 4.1 is similar to the definition given in [1].

Theorem 4.2 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

A curve of osculating order 3 in G is a general helix if and only if
(4.2)τ1=ck1,
where c is a constant.

From (4.2), a curve with k1≠0 is a general helix if and only if (τ1/k1)(s) = constant. As a Euclidean sense, if both k1(s)≠0 and τ1(s) are constants, it is a cylindrical helix. We call such a curve a circular helix.

Theorem 4.3.

Let α be a Frenet curve of osculating order 3. Then α′′(s), α′′′(s), and α′′′′(s) are linearly dependent if and only if α(s) is general helix.

Proof.

If α′′(s), α′′′(s), and α′′′′(s) are linearly dependent, then the following equation holds:
(4.3)|0k10-k12k1′k1τ1-3k1k1′k1′′-k13-k1k22+2k1k2k-2-k1k-222k1′τ1+k1τ1′|=0.
By a direct computation, we have
(4.4)k1τ1′-k1′τ1=0;
it follows that
(4.5)dds(τ1k1)=0.
Thus, τ1/k1 = constant; that is, α is general helix. The converse statement is trivial.

Theorem 4.4.

Let α be a general helix of osculating order 3. Then α is of weak AW(3)-type if and only if α is a circular helix.

Proof.

From (3.7) and (4.2), we can obtain that k1 = constant; it follows that τ1= constant. Thus, α is a circular helix. The converse statement is trivial.

Theorem 4.5.

A general helix of type AW(2) has Frenet curvatures
(4.6)k1(s)=1-(1+c2)s2+d1s+d2,τ1(s)=ck1(s),
where c, d1, and d2 are constants.

Proof.

If α is a general helix of type AW(2), then from (3.9) and (4.2) we have
(4.7)k1′(s)(2k1′(s)τ1(s)+k1(s)τ1′(s))=k1(s)τ1(s)(k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s))=0,(4.8)τ1(s)k1(s)=c;
where c is a constant.

Combining (4.7) and (4.8), we have
(4.9)k1(s)k1′′(s)-3(k1′(s))2-(1+c2)k14(s)=0.
To solve this differential equation, we take
(4.10)k1(s)=x.
Then, (4.9) can be rewritten as the form
(4.11)xd2xds2-3(dxds)2=(1+c2)x4.
Let us put
(4.12)x=yp.
Then (4.11) becomes
(4.13)py2p-1d2yds2-p(2p+1)y2p-2(dyds)2=(1+c2)y4p.
If we choose p=-1/2, then the above equation is
(4.14)d2yds2=-2(1+c2),
its general solution is given by
(4.15)y=-(1+c2)s2+d1s+d2,
where d1 and d2 are constants.

Thus, we have
(4.16)k1(s)=1-(1+c2)s2+d1s+d2,

so, the theorem is proved.

Corollary 4.6.

There exists no a circular helix of osculating order 3 of type AW(2) in G.

Theorem 4.7.

Let α be a general helix of osculating order 3. Then α is of type AW(3) if and only if α is a circular helix.

Proof.

Suppose that α is a general helix of type AW(3). Combining (3.10) and (4.2) we find k13(s)=1, that is, k1(s)=1. From this τ1(s)=c. Thus, α is a circular helix.

Theorem 4.8.

Let α be a curve of osculating order 3. There exists no a general helix of type AW(1).

Proof.

We assume that α is a general helix of type AW(1). Then from (3.8) and (4.2) we have
(4.17)k1′′(s)-k13(s)-k1(s)k22(s)+2k1(s)k2(s)k-2(s)-k1(s)k-22(s)=0,(4.18)k12(s)τ1(s)=c,(4.19)τ1(s)=ck1(s).
From (4.18) and (4.19), we have
(4.20)k1(s)=1.
Thus, (4.17) becomes
(4.21)k22(s)-2k2(s)k-2(s)+k-22(s)=-1,
equivalently to
(4.22)(k2(s)-k-2(s))2=-1.
It is impossible, so the theorem is proved.

Acknowledgments

This paper was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2003994).

BarrosM.General helices and a theorem of LancretŞenolA.YayliY.LC helices in space formsArroyoJ.BarrosM.GarayO. J.Models of relativistic particle with curvature and torsion revisitedBeltranJ. V.MonterdeJ.A characterization of quintic helicesCamcıC.İlarslanK.KulaL.HacısalihoğluH. H.Harmonic curvatures and generalized helices in EnÇiftçiÜ.A generalization of Lancret's theoremFaroukiR. T.HanC. Y.ManniC.SestiniA.Characterization and construction of helical polynomial space curvesArslanK.WestA.Product submanifolds with pointwise 3-planar normal sectionsKülahcıM.BektaşM.ErgütM.On harmonic curvatures of a Frenet curve in Lorentzian spaceÖzgürC.GezginF.On some curves of AW (k)-typeKülahciM.BektaşM.ErgütM.Curves of AW (k)-type in 3-dimensional null conedo Espírito-SantoN.FornariS.FrenselK.RipollJ.Constant mean curvature hypersurfaces in a Lie group with a bi-invariant metricArslanK.ÖzgürC.Curves and surfaces of AW (k) typeStruikD. J.