Abstract

The generalization of Bertrand curves in Galilean 4-space is introduced and the characterization of the generalized Bertrand curves is obtained. Furthermore, it is proved that no special curve is a classical Bertrand curve in Galilean 4-space such that the notion of classical Bertrand curve is definite only in three-dimensional spaces.

1. Introduction

The geometry of curves has long captivated the interests of mathematicians, from the ancient Greeks to the era of Isaac Newton (1643–1727) and the invention of the calculus. It is a branch of geometry that deals with smooth curves in the plane and in the space by methods of differential and integral calculus. The theory of curves is simpler and narrower in scope because a regular curve in Euclidean space has no intrinsic geometry. One of the most important tools used to analyze curve is the Frenet frame, a moving frame that provides a coordinate system at each point of curve that is “best adopted” to the curve near the point.

Bertrand curves discovered by J. Bertrand in 1850 are one of the important and interesting topics of classical special curve theory. A Bertrand curve is defined as a special curve whose principal normal is the principal normal of another curve. It is characterized as curve whose curvature and torsion are in linear relation. There are many works related with Bertrand curves in the Euclidean space and Lorentzian space [1–7].

Galilean 3-space is simply defined as a Klein geometry of the product space whose symmetry group is Galilean transformation group which has an important place in classical and modern physics. A curve in Galilean 3-space is a graph of a plane motion. Note that such a curve is called a world line in 3-dimensional Galilean space. It is well known that the idea of world lines originates in physics and was pioneered by Einstein. The term is now most often in relativity theories, that is, general relativity and special relativity.

From the differential geometric point of view, the study of curves in has its own interest. In recent years, many interesting results on curves in have been obtained by many authors (see [6–10]).

In 4-dimensional Euclidean space, generalized Bertrand curves are defined and characterized by [5]. Moreover, in 4-dimensional semi-Euclidean and also it is proved that there is no timelike curve which is Bertrand curve [4]. Also, generalized Bertrand curves in 5-dimensional Euclidean and Lorentzian space are defined and characterized in [6, 11, 12].

In [8], the author constructed Frenet-Serret frame of a curve in the Galilean 4-space and obtained the mentioned curve’s Frenet-Serret equations.

However, to the best of our knowledge, special Bertrand curves have not been presented in the Galilean 4-space . Thus, the study is proposed to serve such a need. In this regard, we prove that there is no Frenet curve which is a classical Bertrand curve in . We define and characterize -Bertrand curve in four-dimensional Galilean space .

2. Preliminaries

In this section, some fundamental properties of curves in 4D Galilean space are given for the purpose of the requirements [8].

In affine coordinates the Galilean scalar product between two points is defined by

The Galilean cross product in for the vectors , , and is defined by where , , are the standard basis vectors.

The scalar product of two vectors and in is defined by

The norm of vector is defined by See [8].

Let , be a curve parametrized by arclength in . The first vector of the Frenet-Serret frame, that is, the tangent vector of , is defined by Since is a unit vector, we can express Differentiating (7) with respect to , we have

The vector function gives us the rotation measurement of the curve . The real valued function is called the first curvature of the curve . We assume that , for all . Similar to space , the principal vector is defined by in other words See [8].

By the aid of the differentiation of the principal normal vector given in (11), define the second curvature function that is defined by This real valued function is called torsion of the curve . The third vector field, namely, binormal vector field of the curve , is defined by Thus the vector is perpendicular to both and . The fourth unit vector is defined by Here the coefficient is taken to make determinant of the matrix .

The third curvature of the curve by the Galilean inner product is defined by Here, as well known, the set is called the Frenet-Serret apparatus of the curve . We know that the vectors are mutually orthogonal vectors satisfying

For the curve in , we have the following Frenet-Serret equations: See [8].

3. Bertrand Curves in 3-Dimensional Galilean Space

Definition 1. Let and be the curves with ,  , , and for each in and and the Frenet frames in along and , respectively. If is linearly dependent, in other words, if the normal lines of and at are parallel, then a pair of curves is said to be a Bertrand pair in . The curve is called a Bertrand mate of and vice versa. A Frenet framed curve is said to be a Bertrand curve if it admits a Bertrand mate.

Let be Bertrand pair in . Then we can write See [7].

Theorem 2. Let be Bertrand pair in . Then the function defined in the above relation is a constant [7].

Theorem 3. Let be a curve in . Then is a Bertrand curve if and only if is a curve with constant torsion [7].

Definition 4. A -special Frenet-Serret curve in 4-dimensional Galilean space is called a Bertrand curve if there exist a -special Frenet-Serret curve , distinct from , and a regular -map ( for all such that curves and have the same principal normal line at each pair of corresponding points and under . Here and are arclength parameters of and , respectively. In this case, is called a Bertrand mate of and the mate of curves is said to be a Bertrand mate in .

Let be Bertrand mate in . Then we can write

Theorem 5. There is not any Bertrand curve in 4-dimensional Galilean space .

Proof. Let and be the Frenet-Serret frames in along and , respectively. Since is a Bertrand mate, from (19), it holds that By differentiation of (20) with respect to , we obtain From the Frenet-Serret equations, it holds that Since and we get that is, is a constant function on with value (we can use the same letter without confusion). Thus (19) is rewritten as and we get for all . By (25), we can set where is a -function on and Differentiating (26) and using the Frenet-Serret equations, we obtain Since for all , we obtain By , for all and (30), we get that . Thus, by and (27), we obtain that . Therefore, (24) implies that coincides with . This is a contradiction which completes the proof.

4. Special Bertrand Curves in

In this section, we give the notion of special Bertrand curve which is called -Bertrand curve in four-dimensional Galilean space . We obtain a characterization of -Bertrand curve.

Definition 6. Let and be -special Frenet-Serret curves in and a regular -map such that each point of corresponds to the point of for all . Here and are arclength parameters of and , respectively. If the Frenet-Serret -normal plane at each point of coincides with the Frenet-Serret -normal plane at corresponding point of for all , then is called an -Bertrand curve in and is called -Bertrand mate of .

Theorem 7. Let be -special Frenet-Serret curves in with curvature functions , , and . Then is an -Bertrand curve if and only if there exist constant real numbers , , , and satisfying for all .

Proof. We suppose that is an -Bertrand curve parametrized by arclength . Then -Bertrand mate is given by for all , where and are -functions on and is the arclength parameter of . Differentiating (31) with respect to and using the Frenet equations, we obtain for all .
Since the plane spanned by and coincides with the plane spanned by and , we can put and we notice that for all . By the following facts: we get that is, and are constant functions on with values and , respectively. Therefore, for all , (31) is rewritten as and we obtain Here we notice that for all . Thus we can set where is -functions on . Differentiating (39) with respect to and using the Frenet equations, we obtain Since is expressed by linear combination of and , it holds that that is, is a constant function on with value . Thus we obtain for all . Therefore we obtain for all .
If , then it holds that . Thus (43) implies that . Differentiating this equality, we obtain that is, for all . By Theorem 5, this fact is a contradiction. Thus, we must consider only the case of . Then (45) implies Thus, we obtain relation .
The fact that and (46) imply and we obtain for all , where is a constant number. Thus we obtain relation .
Differentiating (43) with respect to and using the Frenet equations, we obtain for all .
From (44) and (45) and , we get for all .
From (38) and (51), it holds that Thus, we obtain By (44) and (45), we can set where for all , where is -functions on .
Differentiating (56) with respect to and using the Frenet equations, we obtain for all . From the above fact, it holds that that is, is a constant function on with value . Thus we obtain Since for all , it holds that
Let be a constant number. Then (51) and (57) imply that is, we obtain relation .
Conversely, we suppose that is a -special Frenet curve in with curvature functions , and satisfying , , and for constant numbers , , , and . Then we define a -curve by for all , where is the arclength parameter of . Differentiating (63) with respect to and using the Frenet equations, we obtain for all . Thus, by relation , we obtain Since relation holds, the curve is a regular curve. Then there exists a regular map defined by where denotes the arclength parameter of , and we obtain for all . Thus the curve is rewritten as for all . Differentiating the above equality with respect to , we obtain
We can define a unit vector field along by for all . By (67) and (69) we obtain for all . Differentiating (70) with respect to and using the Frenet equations, we obtain By the fact that for all , we obtain for all . Then we can define a unit vector field along by for all . Thus we can put where for all . Here is a -function on . Differentiating (74) with respect to and using the Frenet equations, we get Differentiating with respect to , we obtain Differentiating (74) and (75) with respect to and using (78), we get that is, is a constant function on with value . Thus, we find From (74), it holds that Thus we obtain, by (70) and (72), Thus we can define a unit vector field along by for all .
Next we can define a unit vector field along by for all . Now we obtain, by (70), (74), (81), and (83), for all . Thus the frame along is orthonormal and positive. Therefore the curve is a -special Frenet-Serret curve in , and it is obvious that the Frenet-Serret -normal plane at each point of coincides with the Frenet-Serret -normal plane at corresponding point of . Therefore is an -Bertrand curve in .