The Galilean space is considered to be one of the Cayley–Klein spaces, and Roschel was the primary contributor to its development. A Galilean space is the limit case of a pseudo-Euclidean space in which the isotropic cone degenerates to a plane. In this situation, the only shape left is a plane. The limit transition is similar to that encountered when classical mechanics replaced special relativity.

The disadvantage of the Frenet frame is that it is not defined everywhere, namely, if the curve has points where they have zero curvature. At these points, normal and binormal vectors are not defined. Hence, many mathematicians investigated frames that are defined everywhere, even if the curve has zero curvature points. Many frames such as the modified frame, the Bishop frame, the Darboux frame, the equiform frame, and quasi-frame have been investigated and studied in Euclidean space [1–5], Minkowski space [6–11], and Galilean space [12–15].

In Euclidean three-space, the osculating curve is defined as the position vector of the curve residing in the plane consisting of its tangent vector and normal vector. The normal curve is defined as the position vector of the curve residing in the plane consisting of its normal vector and binormal vector. The rectifying curve is defined as the position vector of the curve residing in the plane consisting of its tangent vector and binormal vector. Some studies have been carried out on normal, osculating, and rectifying curves in Euclidean three and four spaces [16–20], Minkowski three and four spaces [21–24], Galilean three and four spaces [12,25–29] and in Sasakian space [30].

In 2015 [1], Dede et al. investigated an alternate adapted frame called the quasi-frame, which followed a space curve, rather than using the Frenet frame. This frame is easier and more accurate than the Frenet frame and the Bishop frame, and it is considered a generalization of the Frenet frame. Many studies have been carried out on the quasi-frame in Euclidean and Minkowski spaces [2,3,31,32]. Furthermore, more recent research studies on position vectors in Galilean three and four spaces were performed with the Frenet frame [33–36].

Rectifying curves, normal curves, and osculating curves are found in the Euclidean space E 3 . These curves meet the fixed point criterion proposed by Cesaro. It is well known that if all the normal planes or osculating planes of a curve in E 3 pass through a given point, then the curve either resides in a sphere or is a planar curve, depending on the two category it falls into. It is also well known that if all rectifying planes of a non-planar curve in E 3 run through a certain point, then the ratio of the curve’s torsion to its curvature is a non-constant linear function. For more details, see [16]. In addition, Ilarslan and Nesovic [17] provided some characterizations for osculating curves in E 3 . They also constructed osculating curves in E 4 as a curve whose position vector always lies in the orthogonal complement of its first binormal vector field. These characterizations were given for osculating curves in E 3 . As a consequence of their findings, they could classify osculating curves according to the curvature functions of those curves and provide both the necessary and sufficient conditions of osculating curves for arbitrary curves in E 4 .

The research is organized as follows: Section 3 introduces the quasi-frame, its relation with the Frenet frame, quasi-formulas, and the quasi-curvatures in Galilean 4-space. Section 4 describes the study of the position vectors in Galilean 4-space. Section 5 characterizes the quasi-rectifying curves. Section 6 introduces and describes the quasi-osculating curves. Section 7 finally proves that there is no normal curve in Galilean 4-space.

In this section, we introduce some basic concepts of Galilean 4-space. The Galilean metric g in Galilean 4-space is defined by g p ,  q = p 1 q 1 , if p 1 ≠ 0  or  q 1 ≠ 0 , p 2 q 2 + p 3 q 3 + p 4 q 4 , if p 1 = 0  and  q 1 = 0 , where p = p 1 ,  p 2 ,  p 3 , p 4 and q = q 1 ,  q 2 ,  q 3 , q 4 . Based on this metric, the Galilean norm of the vector q is given by q = q 1 , if q 1 ≠ 0 , q 2 2 + q 3 2 + q 4 2 , if q 1 = 0 .

In addition, the Galilean cross-product of p , q and s is defined as p × q × s = 0 e 2 e 3 e 4 p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 s 1 s 2 s 3 s 4 , if p 1 ≠ 0  or  q 1 ≠ 0 , e 1 e 2 e 3 e 4 p 1 p 2 p 3 p 4 q 1 q 2 q 3 q 4 s 1 s 2 s 3 s 4 , if p 1 = 0  and  q 1 = 0 , where ( e 1 , e 2 , e 3 , and e 4 ) are the usual bases of R 4 [26,35].

The Galilean G 4 adds even more complexity by investigating all qualities that remain constant despite the spatial motions of objects. It was further clarified that this geometry may be defined as the investigation of properties of 4-dimensional space, the coordinates of which remain unchanged when subjected to a general Galilean transformation [27,29].

A curve in G 4 is a mapping α from an open interval J to G 4 defined as α t = x t , y t , z t , r t , where x ( t ) , y ( t ) , z ( t ) and r ( t ) are differentiable functions. If the curve α is parameterized by the arc length, then it takes the form α s = s , y s , z s , r s .

On the other hand, the Frenet frame in G 4 consists of four orthonormal vectors called the tangent, the principal normal, the first binormal, and the second binormal, and they are denoted, respectively, by T s = α ′ = 1 , y ′ s , z ′ s , r ′ s , N s = 1 κ 1 0 , y ′ , z ′ , r ′ , B 1 s = 1 κ 2 0 , y ′ κ 1 ′ , z ′ κ 1 ′ , r ′ κ 1 ′ , B 2 s = T s × N s × B 1 s , where κ 1 , κ 2 and κ 3 are the first, second, and third Frenet curvatures, respectively. They can be given by κ 1 = y ′ 2 + z ′ 2 , r ′ 2 , κ 2 = g N ′ , N ′ , κ 3 = g B 1 ′ , B 1 ′ . If the Frenet curvatures are constant, then we say the curve is a W-curve.The Frenet formulas of the curve α are T ′ = κ 1 N s , N ′ = κ 2 B 1 s , B 1 ′ = − κ 2 N s + κ 3 B 2 s , B 2 ′ = − κ 3 B 1 s .

Let α ( s ) be a unit speed curve in G 4 . If its position vector always lies in the orthogonal complement of B 1 or B 2 , then a curve α is called an osculating curve in G 4 . If the position vector of α always lies in the orthogonal complement of the normal vector N . Let α ( s ) be an admissible curve in G 4 . We say that α ( s ) is a rectifying curve if the position vector of α always lies in the orthogonal complement of N [26,35].

In this section, we investigate the quasi-frame and its relation with the Frenet frame in G 4 . In addition, quasi-formulas in Galilean 4-space G 4 are investigated. Moreover, the quasi-curvatures are introduced. Let α ( s ) be a curve in G 4 .

The quasi-frame is an alternative to the Frenet frame and involves two fixed unit vectors. We define the quasi frame depending on four orthonormal vectors, T ( s ) called the unit tangent, N q ( s ) called the unit quasi-normal vector, B 1 q ( s ) called the unit first quasi-binormal vector, and B 2 q ( s ) called the unit second quasi-binormal vector. The quasi-frame { T ( s ) , N q ( s ) , B 1 q ( s ) , B 2 q ( s ) } is defined as T = α ′ ‖ α ′ ‖ , N q = T × r 1 × r 2 ‖ T × r 1 × r 2 ‖ , B 2 q = ϵ T × N q × α ‴ ‖ T × N q × α ‴ ‖ and B 1 q = ϵ B 2 q × T × N q , for the projection vectors r 1 and r 2 and ϵ is ± 1 , where the determinant of the matrix is equal to 1. Here, we choose for simple calculations r 1 = ( 0,0,0,1 ) and r 2 = ( 0,0,1,0 ) .

The transformation matrix M keeps the tangent vector T unchanged. Then, we consider three possible planes of rotations. The first rotation M 1 is in the plane spanned by B 1 and B 2 with an angle θ . The second rotation M 2 in the plane is spanned by N and B 2 with an angle ϕ . The third rotation M 3 in the plane is spanned by N and B 1 with an angle ψ as in Figure 1. The quasi-frame can be written in terms of the Frenet frame as M 1 = 1 0 0 0 0 1 0 0 0 0 cos ⁡ θ sin ⁡ θ 0 0 − sin ⁡ θ cos ⁡ θ , M 2 = 1 0 0 0 0 cos ⁡ ϕ 0 sin ⁡ ϕ 0 0 1 0 0 − sin ⁡ ϕ 0 cos ⁡ ϕ , M 3 = 1 0 0 0 0 cos ⁡ ψ sin ⁡ ψ 0 0 − sin ⁡ ψ cos ⁡ ψ 0 0 0 0 1 T N q B 1 q B 2 q = M 1 M 2 M 3 T N B 1 B 2 , The transformation matrix M = M 1 M 2 M 3 can be written as M = 1 0 0 0 0 cos ⁡ θ ⁡ cos ⁡ ψ cos ⁡ θ ⁡ sin ⁡ ψ sin ⁡ ϕ 0 − cos ⁡ θ ⁡ sin ⁡ ψ − cos ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ ϕ cos ⁡ θ ⁡ cos ⁡ ψ − sin ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ ϕ 0 sin ⁡ θ ⁡ sin ⁡ ψ − cos ⁡ ψ ⁡ cos ⁡ θ ⁡ sin ⁡ ϕ − cos ⁡ ψ ⁡ sin ⁡ θ − cos ⁡ θ ⁡ sin ⁡ ψ ⁡ sin ⁡ ϕ 0 0 .

Let the matrix of the quasi-frame be Q and the matrix of the Frenet frame be F . In addition, let the curvature matrix of the quasi-frame be K F and the curvature matrix of the Frenet frame be K Q . Then, we can write Q = T N q B 1 q B 2 q , F = T N B 1 B 2 , K F = 0 κ 1 0 0 0 0 κ 2 0 0 − κ 2 0 κ 3 0 0 − κ 3 0 .

Then, we can write M F = Q , (1) F = M − 1 Q (2) F ′ = K F F , (3) Q ′ = K Q Q . (4) By differentiating Eq. 1 with respect to s , we have M ′ F + M F ′ = Q ′ (5) By substituting Eqs 2–4 into Eq. 5, we have K Q = M ′ + M K f M − 1 . Therefore, K Q = 0 K 1 K 2 K 3 0 0 K 4 K 5 0 − K 4 0 K 6 0 − K 5 − K 6 0 . (6) K 1 = κ 1 ⁡ cos ⁡ ϕ ⁡ cos ⁡ ψ , K 2 = − κ 1 cos ⁡ θ ⁡ sin ⁡ ψ + cos ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ ϕ , K 3 = − κ 1 ⁡ cos ⁡ θ ⁡ cos ⁡ ψ ⁡ sin ⁡ ϕ + κ 1 ⁡ sin ⁡ θ ⁡ sin ⁡ ψ , K 4 = cos ⁡ θ ⁡ cos ⁡ ϕ ψ ′ + cos ⁡ θ κ 2 ⁡ cos ⁡ ϕ − κ 3 ⁡ cos ⁡ ψ ⁡ sin ⁡ ϕ + sin ⁡ θ ϕ ′ + κ 3 ⁡ sin ⁡ ψ , K 5 = − ψ ′ ⁡ cos ⁡ ϕ ⁡ sin ⁡ θ + ϕ ′ ⁡ cos ⁡ θ − κ 2 ⁡ cos ⁡ ϕ ⁡ sin ⁡ θ + κ 3 ⁡ cos ⁡ θ ⁡ sin ⁡ ψ + κ 3 ⁡ cos ⁡ ψ ⁡ sin ⁡ θ ⁡ sin ⁡ ϕ K 6 = θ ′ + ψ ′ ⁡ sin ⁡ ϕ + κ 2 ⁡ sin ⁡ ϕ + κ 3 ⁡ cos ⁡ ϕ ⁡ cos ⁡ ψ

Corollary 3.1

The quasi-frame is considered a generalization to the Frenet frame by putting θ = ϕ = ψ = 0 . In addition, the quasi-formulas are considered generalizations to the Frenet formulas by putting K 1 = κ 1 , K 2 = K 3 = K 5 = 0 , K 4 = κ 2 , K 6 = κ 3 .

Corollary 3.2

The quasi-curvatures K 1 , K 2 , K 3 , K 4 , K 5 , K 6 of the curve are given, respectively, by K 1 = g T ′ , N q , K 2 = g T ′ , B 1 q , K 3 = g T ′ , B 2 q , K 4 = g N q ′ , B 1 q , K 5 = g N q ′ , B 2 q , K 6 = g B 1 q ′ , B 2 q .

In this section, we study the position vectors in G 4 .

We consider a curve in Galilean 4-space G 4 as a curve whose position vector satisfies the parametric equation α s = b 1 s T + b 2 s N q + b 3 s B 1 q + b 4 B 2 q , (7) for some differentiable functions, b i ( s ) and 1 ≤ i ≤ 4 , where T , N q , B 1 q , B 2 q is the quasi-frame. By differentiating Eq. 7 with respect to arclength parameter s and using the quasi Eq. 6, we obtain α ′ s = b 1 ′ T + b 1 K 1 + b 2 ′ − K 4 b 3 − K 5 b 4 N q + b 1 K 2 + b 2 K 4 + b 3 ′ − b 4 K 6 B 1 q + b 1 K 3 + b 2 K 5 + b 3 K 6 + b 4 ′ B 2 q . Hence, b 1 ′ = 0 , b 1 K 1 + b 2 ′ − K 4 b 3 − K 5 b 4 = 0 , b 1 K 2 + b 2 K 4 + b 3 ′ − b 4 K 6 = 0 , b 1 K 3 + b 2 K 5 + b 3 K 6 + b 4 ′ = 0 . Let K 5 = K 6 = 0 and K 1 , K 2 , K 3 , K 4 are constants, so we can find b 1 , b 2 , b 3 as b 1 = s + C , b 2 = C 1 ⁡ cos K 4 s + C 2 ⁡ sin K 4 s − K 2 K 4 s − K 1 K 4 2 + C K 2 K 4 , b 3 = C 3 ⁡ cos K 4 s + C 4 ⁡ sin K 4 s + K 1 K 4 s − K 2 K 4 2 + C K 1 K 4 , b 4 = a s + C . Therefore, we can write completely the curve α s = b 1 s T + b 2 s N q + b 3 s B 1 q + b 4 B 2 q .

In this section, we define the quasi-rectifying curve in the Galilean 4-space and characterize quasi-rectifying curves G 4 .

Definition 1

A curve α ( s ) in the Galilean 4-space is called a quasi-rectifying curve if it has no component in the quasi-normal direction, in other words if g ( α ( s ) , N q ) = 0 . In addition, the curve α ( s ) is called a quasi-rectifying curve if the position vector satisfies the parametric equation α s = a 1 s T + a 2 s B 1 q + a 3 s B 2 q , (8) for some differentiable functions, a i ( s ) and 1 ≤ i ≤ 3 , where T , N q , B 1 q , B 2 q is the quasi-frame.

By differentiating Eq. 8 concerning arclength parameter s and using the quasi Eq. 6, we obtain α ′ s = a 1 ′ T + a 1 K 1 − a 2 K 4 − a 3 K 5 N q + a 1 K 2 + a 2 ′ − a 3 K 6 B 1 q + a 1 K 3 + K 6 a 2 + a 3 ′ B 2 q Hence, a 1 ′ = 1 , (9) / a 2 K 4 + a 3 K 5 = − a 1 K 1 , (10) a 2 ′ − a 3 K 6 = − a 1 K 2 , (11) a 3 ′ + a 2 K 6 = − a 1 K 3 . (12) By solving Eqs 9–12 together, we get a 1 = s + C , a 2 = exp − ∫ K 4 K 6 K 5 d s ∫ exp ∫ K 4 K 6 K 5 d s s + C K 2 − K 1 K 6 K 5 d s , a 3 = exp ∫ K 5 K 6 K 4 ∫ exp − ∫ K 5 K 6 K 4 d s s + C K 1 K 6 K 5 − K 3 d s .

In this section, we define the quasi-osculating curve in the Galilean 4-space and characterize quasi-osculating curves G 4 .

Definition 2

A curve α ( s ) in the Galilean 4-space is called a quasi-osculating curve if it has no component in the first quasi-binormal direction or the second quasi-binormal direction, in other words if g ( α ( s ) , B 1 q ) = 0 or g ( α ( s ) , B 2 q ) = 0 . In addition, the curve α ( s ) is called a quasi-osculating curve if the position vector satisfies the parametric equation α s = μ 1 s T + μ 2 s N q + μ 3 s B 2 q , or α s = λ 1 s T + λ 2 s N q + λ 3 s B 1 q , for some differentiable functions, μ i ( s ) , 0 ≤ i ≤ 3 , λ i ( s ) , and 1 ≤ i ≤ 3 .

In this section, we prove that there is no quasi-normal curve in G 4 .

Definition 3

A curve α ( s ) in the Galilean 4-space is called a quasi-normal curve if it has no component in the tangent direction, in other words if g ( α ( s ) , T ) = 0 . In addition, the curve α ( s ) is called a quasi-normal curve if the position vector satisfies the parametric equation α s = f 1 s N q + f 2 s B 1 q + f 3 s B 2 q , for some differentiable functions, f i ( s ) and 1 ≤ i ≤ 3 , where T , N q , B 1 q , B 2 q is the quasi-frame.

Theorem 7.1

In the Galilean 4-space, there is no quasi-normal curve.

Suppose that α ( s ) = ( s , y ( s ) , z ( s ) , w ( s ) ) is any curve in the Galilean 4-space. Then, the tangent T is given by T = α ′ = 1 , y ′ , z ′ , w ′ . Thus, g α s , T = s ≠ o , ∀ s . Therefore, there is no quasi-normal curve in G 4 .

Corollary 7.1

In the Galilean n -space, there is no normal curve. Therefore, all results in Refs. [27,28] concerning normal curves are not true.

In this study, we investigate the definition of the quasi-frame in Galilean 4-space G 4 and obtain its relation with the Frenet frame in G 4 . In addition, the quasi-formulas and the quasi-curvatures are investigated. Furthermore, the quasi-rectifying curves G 4 and the quasi-osculating curves G 4 are studied according to the quasi-frame in G 4 . Finally, we proved that there is no quasi-normal curve and accordingly normal curve in G 4 .