1 Introduction

The study of curves has a rich history that dates back to ancient Greece, but it has only been in recent times that curves such as slant curves have been studied. Slant curves are curves that do not lie entirely in a plane, but instead are oriented in a particular direction. They have many applications in fields such as physics, engineering, and computer graphics, and their study has posed many challenges due to their complex nature. Helices are an important type of curve in classical differential geometry that are defined by their tangent lines making a fixed angle with the helix axis. According to Lancret’s theorem, a curve is a general helix if the ratio of its curvature and torsion is constant. This theorem, first proposed by M. A. Lancret in 1802 and later demonstrated by B. de Saint Venant in 1845, is a fundamental result in the study of helices in three-dimensional space [22].

In recent years, a generalization of Lancret’s theorem to n-dimensional Riemannian manifolds has been proposed, which uses a Levi-Civita parallel vector field instead of a fixed direction in the standard definition of a general helix [14]. This generalization states that for Riemannian manifolds with general helices, a constant ratio of torsion and curvature is also a defining characteristic in three dimensions. Furthermore, the concept of a slant helix in Euclidean space was introduced by Izumiya and Takeuchi in 2004 [17]. A slant helix is a curve where the principal normal vector makes a constant angle with a fixed direction [1]. Subsequent generalizations of constant angle curves as slant curves of a contact Riemannian 3-manifold were proposed, defining a slant curve as a curve whose tangent vector field and Reeb vector field have a constant angle [4–8,15,16,19].

One particular type of manifold that has gained significant attention in recent times is the doubly warped product manifold. Doubly warped product manifolds are a type of Riemannian manifold that are constructed by taking the product of two Riemannian manifolds and warping them both in a particular way [23]. They have been extensively studied in recent years due to their applications in various fields, including physics, differential geometry, and topology. The use of doubly warped product manifolds and Robertson-Walker space in the analysis of slant curves has been the subject of several studies [3]. In particular, the use of Robertson-Walker space, which are a type of Lorentzian manifold that are commonly used in cosmology, has been shown to be effective in the analysis of slant curves.

In 2022, a research study was carried out by Dursun with the objective of characterizing and classifying slant curves within a three-dimensional Lorentzian warped product space. The investigation was conducted using a rigorous academic and scientific approach as documented in [13]. The main focus of the study was to provide a detailed analysis of the geometric properties of slant curves within this space. The study employed advanced mathematical techniques and analytical tools to deduce the relevant characterizations and classifications of these curves.

Motivated by these recent developments, this paper investigates the use of doubly warped product manifolds and Robertson-Walker space in the analysis of slant curves. Specifically, we focus on the use of Robertson-Walker space to analyze slant curves and classify all slant curves and calculate their curvature and torsion. Additionally, we examine the relationship between doubly warped product manifolds and other types of manifolds, such as warped product manifolds and Minkowski space. The doubly warped product plays a crucial role in differential geometry and mathematical physics, offering a versatile framework for modeling diverse geometric structures. Its significance lies in its ability to generalize classical warped products, enabling the study of curvature properties, Einstein manifolds, and various soliton solutions. This structure finds applications in general relativity, particularly in spacetime modeling, as well as in geometric analysis, where it aids in understanding the interplay between base and fiber manifolds under different warping functions [11,12,21]. Our goal is to provide valuable insights into the use of mathematical structures in the analysis of slant curves and contribute to the ongoing research in this area.

To achieve this goal, we investigate slant curves within the three-dimensional Lorentzian doubly warped product manifold. Our investigation accounts for the causal characteristics of of these curves and how they relate to the overall geometry of the manifold as an application of generalized Robertson-Walker (GRW). We characterize and classify all slant curves in the Lorentzian doubly warped product manifold, and determine their curvature and torsion. Our results provide important insights into the properties of slant curves and their relationship to different types of manifolds. Overall, the study of slant curves and their relationship to doubly warped product manifolds and Robertson-Walker space is an active area of research that continues to yield new insights and discoveries.

2 Preliminaries

This section presents the fundamental formulas for the three-dimensional Lorentzian doubly warped product manifold M, angles between vectors, and Frenet equations.

3 Slant curves in M

In this section, we investigate the properties of slant curves, with a particular emphasis on their characterization. Slant curves are an important class of curves in differential geometry, and understanding their properties is crucial for applications in various fields. Furthermore, we conduct a comprehensive analysis of the curvature and torsion properties of all types of slant curves. The curvature and torsion characteristics of slant curves have significant implications in fields such as computer graphics, robotics, and biophysics [9]. By providing a thorough examination of these properties, we aim to contribute to a better understanding of slant curves.

Lemma 3.1. Let be a unit speed curve in the with two smooth functions f and h defined on I and , respectively. Then

(3.1)

Proof 3.1. Let . Form (2.2a), (2.2b) and (2.2c), we have

(3.2)

as

(3.3)

If η is constant along a unit speed curve in M, then

where is an integration constant and . For simplicity, put (after translation in s), then

(3.4)

In addition, if α is a null curve, we write , there for the slant curve α ( s )  in M is in the form

(3.5)

From now on, and throughout the study unless otherwise stated.

Proposition 3.1. Let M be a manifold, and consider a non-geodesic Frenet curve α in M. The curve can be classified as a slant curve if and only if certain conditions are satisfied along its path. These conditions are as follows

1. If the curve is spacelike, then the equation

(3.6)

holds, where and .

2. If the curve is timelike, then the equation

(3.7)

holds, where and .

Proof 3.2. If we have a non-geodesic Frenet curve α, it is categorized as a slant curve if and only if the condition is satisfied. By using equations (2.5a) and (3.1), we can obtain the following result

(3.8)

Using (3.4) and (3.8), we get (3.6) and (3.7).

Proposition 3.2. Consider a non-geodesic unit speed spacelike curve α in M with a pseudo-orthonormal frame defined along the curve such that . The curve α is classified as a slant curve if and only if the conditions below are satisfied:

(3.9)

Here, is given by sinh ⁡  ϕH ( s ) , and .

Proof 3.3. Consider a non-geodesic unit speed spacelike curve in M that is classified as a slant curve if the condition holds. By using equations (2.6a) and (3.1), we obtain the following expression:

(3.10)

Further, by applying equations (3.4) and (3.10), we arrive at the equation (3.9).

Proposition 3.3. Consider a non-geodesic unit speed null curve α in M with a pseudo-orthonormal Frenet frame defined along the curve. The curve α is classified as a slant curve if and only if the following equation holds true:

(3.11)

Here, is given by  − ηH ( s ) , and .

Proof 3.4. Given a non-geodesic unit speed null curve in M, the curve is classified as a slant curve if and only if the condition holds. By using equations (2.7a) and (3.1), we obtain the expression:

(3.12)

Further, by applying equations (3.5) and (3.12), we arrive at the equation (3.11).

Next, we evaluate the projection of the vector field onto the B component in the Frenet frame  { T , N , B } .

Proposition 3.4. Given a non-geodesic Frenet curve α in M with the Frenet frame  { T , N , B } , the curve is classified as a slant curve if and only if the following equation holds true:

(3.13)

Here, for spacelike curves, , and for timelike curves, , where .

Proof 3.5. Let . Then, using equation (3.8) and considering , we can express as:

(3.14)

Substituting equation (3.14) into the expression for , we obtain:

(3.15)

Using the fact that , equation (3.15) simplifies to equation (3.13).

Proposition 3.5. Consider a non-geodesic unit speed spacelike curve α in M with the Frenet frame , such that . The curve is classified as a slant curve if and only if the following equation holds true:

(3.16)

Here, is given by , where .

Proof 3.6. Let . We can express as follows:

(3.17)

Multiplying equation (3.17) by and using equation (3.9), we obtain:

(3.18)

Using equations (3.4) and (3.18), we can simplify equation (3.18) to obtain equation (3.16).

Proposition 3.6. Consider a non-geodesic unit speed null curve α in M with the Frenet frame . The curve is classified as a slant curve if and only if the following equation holds true:

(3.19)

Here, is given by , where .

Proof 3.7. Let . We can express as follows:

(3.20)

Multiplying equation (3.20) by and using equation (3.11), we obtain:

(3.21)

Using equations (3.20) and (3.21), we can simplify equation (3.21) to obtain equation (3.20).

The following theorems apply to the curve α ( s )  defined on an open interval J ⊂ ℝ in M, for all cases.

Theorem 3.1. Let α ( s )  be a unit-speed slant Frenet curve. Then, the following cases are considered:

1.(a) if α ( s )  is a unit speed spacelike slant Frenet curve, then it is given by

(3.22)

(b) if α ( s )  is unit speed timelike slant Frenet curve, then it is given by

(3.23)

2. if α ( s )  is a null slant curve α ( s ) , then it is given by

(3.24)

for some , and η ≠ 0.

Proof 3.8. Applying equation (3.5) for a unit speed slant Frenet curve , we obtain . Since , we have , which can be rearranged to obtain:

(3.25)

Then,

(3.26)

Using equations (3.4), (3.25), and (3.26), we can obtain equations (3.22) and (3.23). Similarly, for a unit speed slant null curve α, applying equation (3.5) gives , from which we obtain , which can be rearranged to obtain:

(3.27)

and for some , we have

(3.28)

Using equation (3.28) with some smooth function , we obtain equation (3.24).

To facilitate future calculations, we determine the value of the covariant derivative of the tangent vector field T on the surface. Here, T is defined by , where , , and are differentiable functions of the variables t, x, and y. Using equations (2.2a) through (2.2e), we obtain the components of the covariant derivative of T along T as:

(3.29)

where , , and are the second derivatives of , , and , respectively, and and are smooth functions of the coefficients of the second fundamental form of the surface. The expression in equation (3.29) provides the means to compute the covariant derivative of T in the direction of T, which may be useful in future calculations.

Theorem 3.2. 1. For a unit speed spacelike slant Frenet curve α defined by equation (3.22), the curvature function κ and torsion function τ are given by:

(3.30)

where .

(3.31)

where ξ ( N )  and ξ ( B )  are given form (3.4) and (3.6).

2. Let α be a non-geodesic unit speed spacelike curve with a pseudo-orthonormal frame along α, defined by equation (3.22) with . Then, the torsion function τ of α is given by:

(3.32)

Proof 3.9. Let α be unit speed slant spacelike curve. Using (3.29), we obtain from (3.22) that

(3.33)

where and . As α is a Frenet curve, then

(3.34)

From (3.33) and (3.34), we obtain (3.30).

Upon taking the derivative of equation (3.6) using equations (3.1) and (2.5b), we obtain:

(3.35)

By utilizing , , equations (3.4), (3.6), and (3.13), we may derive equation (3.31) from equation (3.35).

For a non-geodesic unit speed spacelike curve α, taking the derivative of equation (3.9) using equations (3.1) and (2.6b), we obtain:

(3.36)

By utilizing equations (3.4) and (3.9), we may derive equation (3.32) from equation (3.36).

Theorem 3.3. Let α be a unit speed timelike slant Frenet curve defined by equation (3.23). Then, the curvature function κ and torsion function τ of α are given by:

(3.37)

where and .

(3.38)

where ξ ( N )  and ξ ( B )  are given form (3.4) and (3.6).

Proof 3.10. A straightforward calculation from (3.32) and (3.29), we get

(3.39)

where . From which and by using (3.4), we obtain (3.37).

Using (2.5b), (3.1) and (3.7), we obtain:

(3.40)

In Equation (3.40), considering , , (3.7) and (3.13), we obtain (3.38).

Theorem 3.4. The torsion function τ of a unit speed null slant curve α is given by:

(3.41)

Proof 3.11. By differentiate (3.11), we get:

(3.42)

In (3.42), by using (3.12) and (3.19), we determine (3.41).

4 Characterization and classification of slant curves in the 3-dimensional Lorentzian warped product

In 2022, Dursun conducted a study of slant curves in the 3-dimensional Lorentzian warped product , characterizing the slant curves and providing a classification of all such curves [13]. Specifically, when the warping function h = 1, several properties of the curve α ( s )  can be derived, including , and , where η = ssinh ⁡  ϕ, for a spacelike curve and η = scosh ⁡  ϕ, for a timelike curve. Under this condition, the manifold becomes equivalent to .

Therefore, several corollaries, depending on Theorems 3.1, 3.2, and 3.3, hold for the curve α ( s )  defined on an open interval J ⊂ ℝ for all s ∈ J in .

Corollary 4.1. If α ( s )  is unit speed spacelike slant Frenet curve, then it is given by

(4.1)

with curvature and torsion are

where , for some , if , if , and is the signature of .

Corollary 4.2. If α ( s )  is unit speed timelike slant Frenet curve, then it is given by

(4.2)

with curvature and torsion are

where , for some and is the signature of .

Corollary 4.3. If α ( s )  is unit speed null slant Frenet curve, then it is given by

(4.3)

with , , and , where C is constant, for some smooth function γ ( s )  on J, and η ≠ 0.

5 Characterization and classification of slant curves in the Minkowski space

In 2011, Barros and others conducted a study on the characterization of general helices in 3-dimensional Minkowski space forms, using the definition of a killing vector field to analyze these curves [2]. In a separate study in 2006, Kosinka and Jüttler provided a proof of Lancret’s theorem for general helices in , contributing to the understanding of the geometric properties of these curves [18].

If we consider the warping functions h = 1 and f = 1, i.e., H ≡ 0 and F ≡ 0, then the Lorentzian doubly warping manifold M is the Minkowski space . Therefore, Theorems 3.1, 3.2 and 3.3 for the curve α ( s )  defined on an open interval J ⊂ ℝ for all s ∈ J in the Minkowski space are

Corollary 5.1. If α ( s )  is unit speed spacelike slant Frenet curve, then it is given by

(5.1)

with curvature and torsion , where , which is general helix. And for some constants m , n ∈ ℝ, α ( s )  becomes a helix if γ ( s ) = ms + n.

Corollary 5.2. If α ( s )  is unit speed timelike slant Frenet curve, then it is given by

(5.2)

with curvature and torsion , where , which is a general helix. And for some constants m , n ∈ ℝ, α ( s )  becomes a helix if γ ( s ) = ms + n.

Corollary 5.3. If α ( s )  is unit speed null slant Frenet curve, then it is given by

(5.3)

with the torsion , where , which is a helix.

6 Conclusion

In conclusion, this study investigated the analysis of slant curves using a mathematical construct known as a doubly warped product manifold, which is frequently employed in the field of three-dimensional Lorentzian geometry. The study classified all slant curves and calculated their curvature and torsion, providing valuable insights into the application of Robertson-Walker space for this purpose. The research also revealed that doubly warped product manifolds have been studied more extensively than other types of manifolds, including warped product manifolds and Minkowski space. These findings contribute to our understanding of the use of mathematical constructs in the analysis of slant curves and serve as a foundation for future research in this area.