Research Article | Open Access
General Helices of AW(k)-Type in the Lie Group
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.
The geometry of curves and surfaces in a 3-dimensional Euclidean space 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 is of constant slop; namely, its tangent makes a constant angle with a fixed direction if and only if the ratio of torsion and curvature is a constant. These curves are said to be general helices. If both and 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, -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 in a 3-dimensional space form is called a general helix if there exists a Killing vector field with constant length along and such that the angle between and is a non-zero constant (see ). A general helix defined by a parallel vector field was studied in . Moreover, in  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 . 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 . Also, in  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 with a bi-invariant metric. Moreover, we characterize general helices of AW(k)-type in the Lie group .
Let be a Lie group with a bi-invariant metric and the Levi-Civita connection of the Lie group . If denotes the Lie algebra of , then we know that is isomorphic to , where is identity of . If is a bi-invariant metric on , then we have for all , , .
Let be a unit speed curve with parameter and an orthonrmal basis of . In this case, we write that any vector fields and along the curve as and , where and are smooth functions. Furthermore, the Lie bracket of two vector fields and is given by Let be the covariant derivative of along the curve , , and , where . Then we have
A curve is called a Frenet curve of osculating order if its derivatives , , ,…, are linearly dependent and are no longer linearly independent for all . To each Frenet curve of order one can associate an orthonormal -frame along (such that ) called the Frenet frame and the functions said to be the Frenet curvatures, such that the Frenet formulas are defined in the usual way: If is a Frenet curve of osculating order in , then we define
Proposition 2.1. Let be a Frenet curve of osculating order 3 in . Then one has
Proof. Let be a Frenet curve of osculating order 3 with the Frenet frame . Since , taking the inner product with , , and , respectively, we have and . Thus, we find From (2.5), we get By using the above similar method, we can obtain and .
Remark 2.2. Let be a 3-dimensional Lie group with a bi-invariant metric. Then it is one of the Lie groups , or a commutative group, and the following statements hold (see [6, 12]).(i)If is , then .(ii)If is , then .(iii)If is a commutative group, then .
Proposition 2.3. Let be a Frenet curve of osculating order 3 in . Then one has where .
Proof. Let be a Frenet curve of osculating order 3 in . Then we have
This implies that
Also, we have the following:
Notation. Let we put
3. Curves of AW(k)-Type
In this section, we consider the properties of curves of AW(k)-type in the Lie group .
Definition 3.1 (see, cf. ). The Frenet curves of osculating order 3 are(i)of type weak AW(2) if they satisfy (ii)of type weak AW(3) if they satisfy where
Definition 3.2 (see ). The Frenet curves of osculating order 3 are(i)of type AW(1) if they satisfy ,(ii)of type AW(2) if they satisfy
(iii)of type AW(3) if they satisfy
From the definitions of type AW(k), we can obtain the following propositions.
Proposition 3.3. Let be a Frenet curve of osculating order 3. Then is of weak -type if and only if
Proposition 3.4. Let be a Frenet curve of osculating order 3. Then is of weak -type if and only if
Proposition 3.5. Let be a Frenet curve of osculating order 3. Then is of -type if and only if where is a constant.
Proposition 3.6. Let be a Frenet curve of osculating order 3. Then is of type if and only if
Proposition 3.7. Let be a Frenet curve of osculating order 3. Then is of type if and only if where 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 with a bi-invariant metric and characterize these curves.
Definition 4.1 (see ). Let be a parameterized curve. Then 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 is parametrized by arc-length, then we have for , where is a constant.
If is a commutative group , then Definition 4.1 reduces to the classical definition (see ). Since a left-invariant vector field in is a Killing vector field, Definition 4.1 is similar to the definition given in .
Theorem 4.2 (see ). A curve of osculating order 3 in is a general helix if and only if where is a constant.
From (4.2), a curve with is a general helix if and only if = constant. As a Euclidean sense, if both and are constants, it is a cylindrical helix. We call such a curve a circular helix.
Theorem 4.3. Let be a Frenet curve of osculating order 3. Then , , and are linearly dependent if and only if is general helix.
Proof. If , , and are linearly dependent, then the following equation holds: By a direct computation, we have it follows that Thus, = constant; that is, is general helix. The converse statement is trivial.
Theorem 4.4. Let be a general helix of osculating order 3. Then is of weak AW(3)-type if and only if is a circular helix.
Theorem 4.5. A general helix of type has Frenet curvatures where , , and are constants.
Proof. If is a general helix of type , then from (3.9) and (4.2) we have
where is a constant.
Combining (4.7) and (4.8), we have To solve this differential equation, we take Then, (4.9) can be rewritten as the form Let us put Then (4.11) becomes If we choose , then the above equation is its general solution is given by where and are constants.
Thus, we have
so, the theorem is proved.
Corollary 4.6. There exists no a circular helix of osculating order 3 of type in .
Theorem 4.7. Let be a general helix of osculating order 3. Then is of type if and only if is a circular helix.
Theorem 4.8. Let be a curve of osculating order 3. There exists no a general helix of type .
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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Copyright © 2012 Dae Won Yoon. 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.