General Helices of AW(k)-Type in the Lie Group

Abstract

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

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Recommended by Hui-Shen Shen

Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 13 October 2012; Accepted 2 December 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The geometry of curves and surfaces in a 3-dimensional Euclidean space ^{... 3} 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 ^{... 3} 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 [variant prime]} 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 ^{... n} . 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 Y9; , YA; and D the Levi-Civita connection of the Lie group G . If ... denotes the Lie algebra of G , then we know that ... is isomorphic to _{T e} G , where e is identity of G . If Y9; , YA; is a bi-invariant metric on G , then we have [figure omitted; refer to PDF] for all X , Y , Z ∈ ... .

Let α :I ⊂ ... [arrow right]G be a unit speed curve with parameter s and { _{V 1} , _{V 2} , ... , _{V n} } an orthonrmal basis of ... . In this case, we write that any vector fields W and Z along the curve α as W = ^{∑ i =1 n} ... _{w i V i} and Z = ^{∑ i =1 n} ... _{z i V i} , where _{w i} :I [arrow right] ... and _{z i} :I [arrow right] ... are smooth functions. Furthermore, the Lie bracket of two vector fields W and Z is given by [figure omitted; refer to PDF] Let _{D ^{α [variant prime]}} W be the covariant derivative of W along the curve α , _{V 1} = ^{α [variant prime]} , and ^{W [variant prime]} = ^{∑ i =1 n} ... ^{w i [variant prime]} _{V i} , where ^{w i [variant prime]} =d _{w i} /ds . Then we have [figure omitted; refer to PDF]

A curve α is called a Frenet curve of osculating order d if its derivatives ^{α [variant prime]} (s ) , ^{α [variant prime][variant prime]} (s ) , ^{α [variant prime][variant prime][variant prime]} (s ) ,..., ^{α (d )} (s ) are linearly dependent and ^{α [variant prime]} (s ) , ^{α [variant prime][variant prime]} (s ) , ^{α [variant prime][variant prime][variant prime]} (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 _{V 1} (s ) , _{V 2} (s ) , _{V 3} (s ) , ... , _{V d} (s ) along α (such that ^{α [variant prime]} (s ) = _{V 1} (s ) ) called the Frenet frame and the functions _{k 1} , _{k 2} , ... , _{k d -1} :I [arrow right] ... said to be the Frenet curvatures, such that the Frenet formulas are defined in the usual way: [figure omitted; refer to PDF] If α :I [arrow right]G is a Frenet curve of osculating order 3 in G , then we define [figure omitted; refer to PDF]

Proposition 2.1.

Let α be a Frenet curve of osculating order 3 in G . Then one has [figure omitted; refer to PDF]

Proof.

Let α be a Frenet curve of osculating order 3 with the Frenet frame { _{V 1} , _{V 2} , _{V 3} } . Since [ _{V 1} , _{V 2} ] = _{a 1 V 1} + _{a 2 V 2} + _{a 3 V 3} , taking the inner product with _{V 1} , _{V 2} , and _{V 3} , respectively, we have _{a 1} = _{a 2} =0 and Y9; [ _{V 1} , _{V 2} ] , _{V 3} YA; = _{a 3} . Thus, we find [figure omitted; refer to PDF] From ( 2.5), we get [figure omitted; refer to PDF] By using the above similar method, we can obtain [ _{V 1} , _{V 3} ] = -2 _{k - 2 V 2} and [ _{V 2} , _{V 3} ] =2 _{k - 2 V 1} .

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 ) , ^{S 3} or a commutative group, and the following statements hold (see [ 6, 12]).

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

(ii) If G is ^{S 3} [congruent with]SU (2 ) , then _{k - 2} (s ) =1 .

(iii): 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 [figure omitted; refer to PDF] where _{τ 1} (s ) = _{k 2} (s ) - _{k - 2} (s ) .

Proof.

Let α be a Frenet curve of osculating order 3 in G . Then we have [figure omitted; refer to PDF] This implies that [figure omitted; refer to PDF] Also, we have the following: [figure omitted; refer to PDF]

Notation. Let we put [figure omitted; refer to PDF]

3. Curves of AW(k)-Type

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

Definition 3.1 (see, cf. [ 13]).

The Frenet curves of osculating order 3 are

(i) of type weak AW(2) if they satisfy [figure omitted; refer to PDF]

(ii) of type weak AW(3) if they satisfy [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Definition 3.2 (see [ 8]).

The Frenet curves of osculating order 3 are

(i) of type AW(1) if they satisfy _{N 3} (s ) =0 ,

(ii) of type AW(2) if they satisfy [figure omitted; refer to PDF]

(iii): of type AW(3) if they satisfy [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF]

Proposition 3.5.

Let α be a Frenet curve of osculating order 3. Then α is of AW (1 ) -type if and only if [figure omitted; refer to PDF] 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 [figure omitted; refer to PDF]

Proposition 3.7.

Let α be a Frenet curve of osculating order 3. Then α is of type AW (3 ) if and only if [figure omitted; refer to PDF] 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 [ 6]).

Let α :I [arrow right]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-length s , then we have [figure omitted; refer to PDF] for X ∈ ... , where θ is a constant.

If G is a commutative group ^{... 3} , then Definition 4.1reduces to the classical definition (see [ 14]). Since a left-invariant vector field in G is a Killing vector field, Definition 4.1is similar to the definition given in [ 1].

Theorem 4.2 (see [ 6]).

A curve of osculating order 3 in G is a general helix if and only if [figure omitted; refer to PDF] where c is a constant.

From ( 4.2), a curve with _{k 1} ...0;0 is a general helix if and only if ( _{τ 1} / _{k 1} ) (s ) = constant. As a Euclidean sense, if both _{k 1} (s ) ...0;0 and _{τ 1} (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 [variant prime][variant prime]} ( s ) , ^{α [variant prime][variant prime][variant prime]} ( s ) , and ^{α [variant prime][variant prime][variant prime][variant prime]} ( s ) are linearly dependent if and only if α (s ) is general helix.

Proof.

If ^{α [variant prime][variant prime]} ( s ) , ^{α [variant prime][variant prime][variant prime]} ( s ) , and ^{α [variant prime][variant prime][variant prime][variant prime]} ( s ) are linearly dependent, then the following equation holds: [figure omitted; refer to PDF] By a direct computation, we have [figure omitted; refer to PDF] it follows that [figure omitted; refer to PDF] Thus, _{τ 1} / _{k 1} = 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 _{k 1} = constant; it follows that _{τ 1} = 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 [figure omitted; refer to PDF] where c , _{d 1} , and _{d 2} are constants.

Proof.

If α is a general helix of type AW (2 ) , then from ( 3.9) and ( 4.2) we have [figure omitted; refer to PDF] where c is a constant.

Combining ( 4.7) and ( 4.8), we have [figure omitted; refer to PDF] To solve this differential equation, we take [figure omitted; refer to PDF] Then, ( 4.9) can be rewritten as the form [figure omitted; refer to PDF] Let us put [figure omitted; refer to PDF] Then ( 4.11) becomes [figure omitted; refer to PDF] If we choose p = -1 /2 , then the above equation is [figure omitted; refer to PDF] its general solution is given by [figure omitted; refer to PDF] where _{d 1} and _{d 2} are constants.

Thus, we have [figure omitted; refer to PDF]

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 ^{k 1 3} (s ) =1 , that is, _{k 1} (s ) =1 . From this _{τ 1} (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 [figure omitted; refer to PDF] From ( 4.18) and ( 4.19), we have [figure omitted; refer to PDF] Thus, ( 4.17) becomes [figure omitted; refer to PDF] equivalently to [figure omitted; refer to PDF] 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).

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