Processing math: 100%
Research article Special Issues

Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space

  • The main goal of this paper is to investigate the evolution equations for special types of timelike ruled surfaces with significant geometric and physical applications in Lorentz-Minkowski 3-space E31. Using the alternative frame associated with the basic curve of these surfaces, we explored their key geometric properties. Our analysis provided insights into the dynamics of local curvatures during their evolutions, enhancing the understanding of surface behavior. Finally, we present applications of our preliminary findings that contribute to the broader field of differential geometry.

    Citation: Yanlin Li, H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, M. Khalifa Saad. Geometric visualization of evolved ruled surfaces via alternative frame in Lorentz-Minkowski 3-space[J]. AIMS Mathematics, 2024, 9(9): 25619-25635. doi: 10.3934/math.20241251

    Related Papers:

    [1] Wei Zhang, Pengcheng Li, Donghe Pei . Circular evolutes and involutes of spacelike framed curves and their duality relations in Minkowski 3-space. AIMS Mathematics, 2024, 9(3): 5688-5707. doi: 10.3934/math.2024276
    [2] Emad Solouma, Ibrahim Al-Dayel, Meraj Ali Khan, Youssef A. A. Lazer . Characterization of imbricate-ruled surfaces via rotation-minimizing Darboux frame in Minkowski 3-space E31. AIMS Mathematics, 2024, 9(5): 13028-13042. doi: 10.3934/math.2024635
    [3] Ayman Elsharkawy, Clemente Cesarano, Abdelrhman Tawfiq, Abdul Aziz Ismail . The non-linear Schrödinger equation associated with the soliton surfaces in Minkowski 3-space. AIMS Mathematics, 2022, 7(10): 17879-17893. doi: 10.3934/math.2022985
    [4] Chang Sun, Kaixin Yao, Donghe Pei . Special non-lightlike ruled surfaces in Minkowski 3-space. AIMS Mathematics, 2023, 8(11): 26600-26613. doi: 10.3934/math.20231360
    [5] Emad Solouma, Mohamed Abdelkawy . Family of ruled surfaces generated by equiform Bishop spherical image in Minkowski 3-space. AIMS Mathematics, 2023, 8(2): 4372-4389. doi: 10.3934/math.2023218
    [6] Kemal Eren, Hidayet Huda Kosal . Evolution of space curves and the special ruled surfaces with modified orthogonal frame. AIMS Mathematics, 2020, 5(3): 2027-2039. doi: 10.3934/math.2020134
    [7] Cai-Yun Li, Chun-Gang Zhu . Construction of the spacelike constant angle surface family in Minkowski 3-space. AIMS Mathematics, 2020, 5(6): 6341-6354. doi: 10.3934/math.2020408
    [8] Yanlin Li, Kemal Eren, Soley Ersoy . On simultaneous characterizations of partner-ruled surfaces in Minkowski 3-space. AIMS Mathematics, 2023, 8(9): 22256-22273. doi: 10.3934/math.20231135
    [9] Samah Gaber, Abeer Al Elaiw . Evolution of null Cartan and pseudo null curves via the Bishop frame in Minkowski space R2,1. AIMS Mathematics, 2025, 10(2): 3691-3709. doi: 10.3934/math.2025171
    [10] Ayman Elsharkawy, Ahmer Ali, Muhammad Hanif, Fatimah Alghamdi . Exploring quaternionic Bertrand curves: involutes and evolutes in E4. AIMS Mathematics, 2025, 10(3): 4598-4619. doi: 10.3934/math.2025213
  • The main goal of this paper is to investigate the evolution equations for special types of timelike ruled surfaces with significant geometric and physical applications in Lorentz-Minkowski 3-space E31. Using the alternative frame associated with the basic curve of these surfaces, we explored their key geometric properties. Our analysis provided insights into the dynamics of local curvatures during their evolutions, enhancing the understanding of surface behavior. Finally, we present applications of our preliminary findings that contribute to the broader field of differential geometry.



    The investigation of surfaces and alternative frames in Lorentz–Minkowski space offers insights into the interplay between differential geometry and special relativity [1,2,3,4].

    Ruled surfaces are surfaces that can be generated by moving a straight line in space according to specific rules. These surfaces have applications in various fields, including architecture, computer graphics, and differential geometry itself. While ruled surfaces are often studied in Euclidean or Riemannian geometries, examining them in the context of E31 opens new perspectives and challenges due to the Lorentzian nature of spacetime. The use of alternative frames in differential geometry [5,6] provides a basis for understanding the geometric and physical properties of spacetime. This approach, extensively discussed in Wald's textbook [7], allows for the decomposition of the metric tensor into components corresponding to different types of vectors in the frame. Alternative frames have proven valuable in studying the kinematics of particles and observers in curved spacetimes, enabling the interpretation of relativistic effects and gravitational interactions. Furthermore, in the fields of partial differential equations, geometric analysis, mathematical physics, etc., soliton theory is a critical theory that attracts the attention of many researchers [8,9,10,11]. There are many applications of solitons in applied mathematics and pure mathematics, especially in partial differential equations, ordinary partial differential equations, Lie algebras, Lie groups, differential geometry, and algebraic geometry [12,13,14,15]. In [16], Manukure and Booker presented an overview of solitons and applications; they stated the research history of solitons and showed further developments. The paper [16] shows that solitons appear in various fields, particularly in physical contexts, including fluid dynamics, optical fibers, and quantum field theory. In each case, they are described by specific evolution equations that capture the relevant physics [17]. This work aims to explore the evolution of different types of timelike ruled surfaces using alternative frames in Minkowski space. By investigating the interplay between these surfaces and alternative frames, we enhance our understanding of the geometric structures of these surfaces in Lorentz-Minkowski 3-space.

    This paper is organized as follows: In Section 2, we provide a brief overview of curves and timelike ruled surfaces and explore the relationship between these surfaces and alternative frames in Lorentz–Minkowski 3-space. Section 3 presents evolution equations for a given timelike curve and some timelike ruled surfaces generated by the alternative frame vectors of their associated timelike curves in Lorentz–Minkowski 3-space. Illustrated examples to support our main results are provided in Section 4, and we conclude with a summary of our findings in Section 5.

    Lorentz–Minkowski 3-space E31 is the real vector space E3 augmented by the Lorentzian inner product

    a,bE31=a1b1+a2b2+a3b3, (2.1)

    where a=(a1,a2,a3) and b=(b1,b2,b3)E31. The norm of b is b=b,bE31.

    The cross product of a and b is given by

    aE31b=a3b2+a2b3,a3b1+a1b3,a1b2a2b1. (2.2)

    If r(s):IRE31 is a regular curve described by

    r(s)=(y(s),z(s),w(s)), (2.3)

    where I is an open interval and y(s),z(s), and w(s)C3, then r is spacelike if r(s),r(s)E31>0, timelike if r(s),r(s)E31<0, and lightlike if r(s),r(s)E31=0, for all sI.

    The derivatives of the Frenet frame vectors of r(s), which has tangent T(s), principal normal n(s), and binormal p(s), take the form:

    s(T(s)n(s)p(s))=(0κ(s)0ϵpκ(s)0τ(s)0ϵTτ(s)0)(T(s)n(s)p(s)), (2.4)

    where

    T,TE31=ϵT,n,nE31=ϵn,andp,pE31=ϵTϵn=ϵp, (2.5)

    and

    T×E31n=p,n×E31p=ϵnT,andp×E31T=ϵTn. (2.6)

    The functions κ(s) and τ(s) represent the curvature and the torsion of the curve, respectively. For more details, see [2]. Additionally, this frame satisfies the following conditions:

    T(s)=dr/ds||dr/ds||,n(s)=Ts(s)||Ts(s)||,p(s)=T(s)E31n(s),

    where Ts(s)=dT(s)/ds.

    In this context, the time evolution equations for the given curve are written as follows:

    t(Tnp)=(0αβα0γβγ0)(Tnp),

    where α, β, and γ are the velocities of the curve, dependent on its curvatures.

    Next, we explore the alternative frame of r(s) as a timelike curve in Lorentz–Minkowski 3-space. By examining this frame, we gain insights into the geometric properties and physical aspects of the curve, offering a fresh perspective on its behavior within the context of differential geometry.

    Consider r=r(s) as a given timelike curve with arc length parameter s in E31. Let {T(s),n(s),p(s)} and {n(s),C(s),W(s)} be the Frenet frame and the alternative frame of r, respectively. The alternative frame can then be defined as follows:

    C(s)=n(s)||n(s)||, W(s)=τ(s)|κ2(s)τ2(s)|T(s)+κ(s)|κ2(s)τ2(s)|(s)p(s), (2.7)

    where n, C, and W represent the spacelike normal, the timelike derivative of the normal, and the spacelike Darboux vector of r(s), respectively [5]. Clearly, the Darboux vector W is orthogonal to the normal vector n(s) [6,18].

    The derivative formulas for the alternative frame of r(s) can be expressed as follows:

    s(n(s)C(s)W(s))=(0σ(s)0σ(s)0ϱ(s)0ϱ(s)0)(n(s)C(s)W(s)), (2.8)

    where

    σ(s)=|κ2(s)τ2(s)|,ϱ(s)=κ2(s)(τ(s)κ(s))κ2(s)τ2(s), (2.9)

    are differentiable functions referred to as alternative curvatures that depend on the curvatures of r(s).

    Since the principal normal vector n(s) is common to both frames, the relationship between the two frames can be expressed in matrix form as [5]:

    (Tnp)=(0κ(s)|κ2(s)τ2(s)|τ(s)|κ2(s)τ2(s)|1000τ(s)|κ2(s)τ2(s)|κ(s)|κ2(s)τ2(s)|)(nCW), (2.10)

    or

    (nCW)=(010κ(s)|κ2(s)τ2(s)|0τ(s)|κ2(s)τ2(s)|τ(s)|κ2(s)τ2(s)|0κ(s)|κ2(s)τ2(s)|)(Tnp). (2.11)

    Considering that

    n×E31C=W,C×E31W=n,andn×E31W=C, (2.12)

    and to simplify the form of our equations, we will use the following symbols:

    ϝ1=κ(s)|κ2(s)τ2(s)|,ϝ2=τ(s)|κ2(s)τ2(s)|.

    The equations of motion for the alternative frame {n(s),C(s),W(s)} of r(s) can be expressed broadly, similar to Eq (2.8) as [19]:

    t(n(s)C(s)W(s))=(0δ(s)θ(s)δ(s)0ϕ(s)θϕ(s)0)(n(s)C(s)W(s)), (2.13)

    where δ, θ, and ϕ represent the alternative velocities of the aforementioned curve.

    The alternative frame provides a distinct basis for analyzing the geometry of curves and surfaces. The vectors in this alternative frame often offer insights into various geometric properties and can simplify the analysis of complex structures [20]. In [21], a novel class of ruled surface known as the C-ruled surface is introduced, defined through the alternative frame associated with a base curve. The differential geometric properties of this surface are examined, including the striction line, distribution parameter, fundamental forms, as well as Gaussian and mean curvatures. In our work, we will examine the evolution equations for specific types of ruled surfaces with significant geometric applications. By employing the alternative frame associated with the basic curve of these surfaces, we will investigate their key geometric properties. This comprehensive analysis will provide deeper insights into the dynamics of the local curvatures during their evolution. Our work aims to advance the understanding of surface behavior in Lorentz–Minkowski 3-space, with implications across various disciplines.

    A ruled surface is one generated by moving a straight line, known as the ruling line, according to specific rules. Each ruling line lies entirely on the surface, forming a family of lines that cover the surface. Mathematically, a ruled surface can be defined as follows (see [3,22,23,24] for more details):

    S(s,v)=r(s)+vΦ(s)|sI,vJ, (2.14)

    where r(s) is the base curve of S(s,v) in Lorentz–Minkowski 3-space, v is a function defined on an interval J, and Φ is a fixed vector representing the direction of the ruling lines. The choice of the curve r(s) and the function v determines the specific properties and shape of the ruled surface.

    Definition 2.1. A surface in Lorentz–Minkowski 3-space is classified as spacelike or timelike based on the nature of the induced metric at the surface: a positive definite Riemannian metric corresponds to a spacelike surface, while a negative definite Riemannian metric corresponds to a timelike surface. Alternatively, a spacelike surface has a normal vector that is timelike, whereas a timelike surface has a normal vector that is spacelike[25].

    For the ruled surface defined by Eq (2.14), the normal vector at a point is defined as a vector perpendicular to the tangent plane of the surface at that point. This normal vector is crucial for defining the geometry of the surface and is denoted by

    N=Ss×Sv||Ss×Sv||. (2.15)

    The first and second fundamental forms on the surface S, along with their quantities, are respectively expressed by

    I=dS,dS=Eds2+2Fdsdv+Gdv2, (2.16)

    where,

    E=Ss,Ss,F=Ss,Sv=Sv,Ss,G=Sv,Sv,

    and

    II=dS,N=eds2+2fdsdv+gdv2, (2.17)

    noting that

    e=Sss,N,f=Ssv,N=Svs,N,g=Svv,N.

    The Gaussian and mean curvatures of S play a vital role in characterizing the shape and properties of S. They are given by the following forms:

    K=ϵNegf2EGF2=Det(h)Det(Δ);ϵN=N,N, (2.18)

    and

    H=ϵN2eG2fF+gE(EGF2)=12ϵNtr(hΔ), (2.19)

    where Δ=EGF2, h=egf2, and h denotes the inverse matrix of h [25].

    It is worth noting that, as stated in [26], the surface evolution denoted by S(s,v,t) and its corresponding flow S(s,v,t)t are considered inextensible if the following condition is satisfied:

    {Et=0,Ft=0,Gt=0. (2.20)

    In this section, we focus on deriving the evolution equations for special types of ruled surfaces using an alternative frame for their curves. To achieve this, we first derive the evolution equations for a timelike curve, highlighting its unique geometric properties and implications within differential geometry. The main result of this analysis is presented in the following theorem.

    Theorem 3.1. Let r=r(s,t) be a given timelike curve which has the alternative frame {n,C,W} in E31, then the evolution equations of the alternative curvatures of r can be described as:

    {σt=δs+ϱθ,ϱt=ϕsσθ, (3.1)

    where δ, ϕ and θ are the velocities of r.

    Proof. Let us write Eq (2.8) in a simple form as

    As=LA, (3.2)

    where

    A=(nCW),L=(0σ(s)0σ(s)0ϱ(s)0ϱ(s)0).

    Similar to that procedure, Eq (2.13) can be reformulated as

    At=MA, (3.3)

    where

    M=(0δ(s)θ(s)δ(s)0ϕ(s)θϕ(s)0).

    From this point, by applying the compatibility conditions sAt=tAs and making some calculations, one can obtain

    LtMs+[L,M]=03×3, (3.4)

    with Lie bracket [L,M]=LMML.

    We conclude from Eq (3.4), after simple calculations, we obtain the following system of equations

    {σtδsϱθ=0,ϱtϕs+σθ=0,θsσϕ+ϱδ=0,

    which leads to the completeness of the proof.

    Our focus now shifts to deriving evolution equations for specific ruled surfaces generated by the alternative frame vectors associated with their curves. This will be addressed through the following theorems.

    Theorem 3.2. Consider r=r(s,t) is a timelike curve that has the alternative frame {n,C,W} in Lorentz–Minkowski 3-space. Let Sn be a timelike n-ruled surface whose r is its base curve, then the following are hold:

    1) The evolution equation of Sn is

    ϝ2ϝ2t(ϝ1+vσ)ϝ1t=0. (3.5)

    2) Sn is minimal surface if and only if

    ϝ2τ2(ϝ1+vτ1)s=0.

    3) Sn is developable if and only if

    (ϝ1+vσ)ϝ2ϱ=0.

    Proof. Since, the ruled surface Sn can be written as:

    Sn(s,v,t)=r(s,t)+vn(s,t), (3.6)

    then by differentiating this equation, we get

    {(Sn)s=(ϝ1+vσ)CϱW,(Sn)v=n.

    From Eq (2.15), the normal on N is obtained:

    N=1τ2κ2(s)τ2(s)+(ϝ1+vσ)2(0,ϝ2,ϝ1vσ). (3.7)

    In the light of the above, the first fundamental quantities are

    E=ϝ22(ϝ1+vσ)2,F=0,G=1, (3.8)

    which lead to

    I=EGF2=ϝ22(ϝ1+vσ)2. (3.9)

    The second derivatives of Eq (3.6) with respect to s and v are expressed as

    {(Sn)ss=σ(ϝ1+vσ)n+((ϝ1+vσ)sϝ2ϱ)C+(ϱ(ϝ1+vσ)(ϝ2)s)W,(Sn)vv=0,(Sn)vs=(Sn)sv=σC.

    From this, the second fundamental form is given as follows:

    II=egf2=(σ2((ϝ1+vσ)ϝ2ϱ)2(ϝ1+vσ)2ϝ22), (3.10)

    where,

    {e=(Sn)ss,N=ϝ22ϱϝ2(ϝ1+vσ)s(ϝ1+vσ)2ϝ22,f=(Sn)sv,N=(Sn)vs,N=σ((ϝ1+vσ)ϝ2ϱ)(ϝ1+vσ)2ϝ22,g=(Sn)vv,N=0. (3.11)

    From the aforementioned data, the Gaussian and mean curvatures for the surface Sn are given as follows:

    {K=(σ((ϝ1+vσ)ϝ2ϱ)ϝ22+(ϝ1+vσ)2)2,H=12(ϝ22ϱϝ2(ϝ1+vσ)s(ϝ22+(ϝ1+vσ)32), (3.12)

    therefore, the surface Sn is minimal if and only if

    ϝ22ϱϝ2(ϝ1+vσ)s=0.

    On the other hand, the surface Sn can be described as a developable if and only if

    σ((ϝ1+vσ)ϝ2ϱ)=0.

    In light of the benefit of Eq (2.20), the evolution of the surface is obtained:

    ϝ2ϝ2t(ϝ1+vσ)ϝ1t=0.

    This finishes the proof.

    Theorem 3.3. Assume that SC is a timelike C-ruled surface and r(s,t) be its timelike curve, which has the alternative frame vectors n,C,W in Lorentz–Minkowski 3-space. The surface SC satisfies the following:

    1) It is minimal if and only if

    (vσ(ϝ1(σϱ)(vσ)s+(ϝ1σ+vϱ)s)+ϝ2(ϝ1σ+(vσ)s)+ϝ1σ(ϝ2+v(σ+ϱ)))=0 (3.13)

    holds.

    2) The surface is developable if and only if

    vσ+vϱϝ2=0 (3.14)

    is hold.

    3) It has the evolution equation:

    ϝ1ϝ1t+(ϝ2+vϱ)ϝ2t=0. (3.15)

    Proof. The parametric representation of the SC surface can be formulated as

    SC(s,v,t)=r(s,t)+vC(s,t). (3.16)

    The partial differentials of the surface SC with respect to s and v are given from

    {(SC)s=vσn+ϝ1C+(ϝ2+vϱ)W,(SC)v=C.

    Again, the surface differentials with respect to the parameters s and v are expressed as follows:

    {(SC)ss=(σϝ1+(vσ)s)n,+(vσ2+(ϝ1)s+ϱ(ϝ2+vϱ))C+((ϝ2+vϱ)s+ϝ1ϱ)W(SC)vv=0,(SC)vs=(SC)sv=σn+ϱW.

    From this, the normal relative to the surface is given by

    N=1(vσ)2+(ϝ2+vϱ)2(ϝ2vσ,0,vσ). (3.17)

    In light of the above-mentioned data related to surface calculations, the first and second fundamental forms with their quantities are, respectively

    E=(vσ)2ϝ21+(ϝ2+vϱ)2,F=ϝ1,G=1, (3.18)

    which give us

    I=(vσ)2(ϝ2+vϱ)2, (3.19)

    and,

    {e=vσ(ϝ1(σϱ)(vσ)s+(ϝ1σ+vϱ)s)+ϝ2(ϝ1σ+vϱ)s)(vσ)2+(ϝ2+vϱ)2,f=σ(ϝ2+vσ+vϱ)(vσ)2+(ϝ2+vϱ)2,g=0. (3.20)

    From this, we have

    II=σ2(ϝ2+vσ+vϱ)2(vσ)2+(ϝ2+vϱ)2. (3.21)

    The geometric meanings of the surface SC are represented by its Gaussian and mean curvatures. They are given from

    {K=(σ(ϝ2+vσ+vϱ)(vσ)2+(ϝ2+vϱ)2)2,H=12vσ(ϝ1(σϱ)(vσ)s+(ϝ1σ+vϱ)s)+ϝ2(ϝ1σ+(vσ)s)+ϝ1σ(ϝ2+v(σ+ϱ))((vσ)2+(ϝ2+vϱ)2)32. (3.22)

    As a result of the above, the surface SC is minimal when

    (vσ(ϝ1(σϱ)(vσ)s+(ϝ1σ+vϱ)s)+ϝ2(ϝ1σ+(vσ)s)+ϝ1σ(ϝ2+v(σ+ϱ)))=0

    holds.

    Besides, the surface SC is classified as a developable whenever

    ϝ2+vσ+vϱ=0,

    is hold.

    By applying Eq (2.20), the evolution condition for this surface is expressed as

    ϝ1ϝ1t+(ϝ2+vϱ)ϝ2t=0.

    Hence, the proof is completed.

    Theorem 3.4. Suppose that SW is a timelike W-ruled surface generated by the alternative vector W of its base curve r in Lorentz–Minkowski 3-space. Then, the following statements are satisfied:

    1) SW is a developable surface.

    2) It has the following evolution equation:

    ϝ2ϝ2t(ϝ1+vϱ)ϝ1t=0. (3.23)

    Proof. Since the surface SW has the following parametric representation

    SW(s,v,t)=r(s,t)+vW(s,t), (3.24)

    then by differentiating this equation twice with respect to s and v, we obtain the following

    {(SW)s=(ϝ1+vϱ)Cϝ2W,(SW)v=σW,(SW)ss=σ(ϝ1+vϱ)n+(ϝ2ϱ+(ϝ1+vϱ)s)C+(ϱ(ϝ1+vϱ)(ϝ2)s)W,(SW)vv=0,(SW)vs=(SW)sv=ϱC.

    From this, we have the normal

    N=(1,0,0), (3.25)

    and

    {E=(SW)s,(SW)s=(ϝ1+vϱ)2+ϝ22,F=(SW)s,(SW)v=ϝ2,G=(SW)v,(SW)v=1, (3.26)
    I=(ϝ1+vϱ)2, (3.27)

    and also

    e=(SW)ss,N=σ(ϝ1+vϱ),f=(SW)vs,N=0,g=(SW)vv,N=0, (3.28)
    II=0. (3.29)

    Further, the Gaussian and mean curvatures of SW are calculated as

    K=0,H=σ2(ϝ1+vϱ), (3.30)

    As a consequence, this surface is developable and not minimal.

    Under the previous data, the evolution equation for the surface SW is given by

    ϝ2ϝ2t(ϝ1+vϱ)ϝ1t=0.

    Thus, the result is clear.

    In this section, we are interested in providing a practical example to demonstrate the theoretical results that we obtained through our study of the three special ruled surfaces: n-ruled, C-ruled, and W-ruled surfaces.

    Consider r(s,t) be a timelike curve given by (see Figure 1)

    r(s,t)=(t sinh2s,t cosh2s,t s). (4.1)
    Figure 1.  The timelike curve r.

    The tangent T, the normal n and the binormal p of r are, respectively

    {T=(2cosh2s,2sinh2s,1),n=(sinh2s,cosh2s,0),p=(cosh2s,sinh2s,2).

    The curvature functions of the considered curve are

    κ=2t,τ=2t,

    also, the alternative frame vectors n,C,W of r are given as follows:

    {n=(sinh2s,cosh2s,0),C=(cosh2s,sinh2s,0),W=(0,0,1). (4.2)

    From this, the alternative curvatures of r are calculated as

    σ=2t,ϱ=0.

    The n-ruled surface that has r as a base curve is expressed by the following representation (see Figure 2)

    Sn=(tsinh(2s)+vsinh(2s),tcosh(2s)vcosh(2s),ts), (4.3)
    Figure 2.  The evolved n-ruled surface Sn of the curve r.

    which has the normal

    N=(tcosh(2s)2v(v+2tcosh(22s))+t2cosh(42s),tsinh(2s)2v(v+2tcosh(22s))+t2cosh(42s),v+tcosh(22s)v2+2tvcosh(22s)+12t2cosh(42s)).

    The first and second fundamental coefficients of the surface can be calculated, respectively

    E=t22v24tvcosh(22s), F=2tsinh(22s),G=1,

    and

    {e=2t2sinh(22s)2v(v+2tcosh(22s))+t2cosh(42s),f=tv2+2tvcosh(22s)+12t2cosh(42s),g=0.

    Also, the surface's curvatures are

    K=2t2(2v2+4tvcosh(22s)+t2cosh(42s))2,H=t2(2+t2+2v2+4tvcosh(22s))sinh(22s)(2v2+4tvcosh(22s)+t2cosh(42s))3/2. (4.4)

    Similarly, the ruled surface that is generated by the vector C of the alternative frame of r can be written as (see Figure 3)

    SC=(vcosh(2s)+tsinh(2s),tcosh(2s)+vsinh(2s),ts). (4.5)
    Figure 3.  The evolved C-ruled surface SC of the curve r.

    Straightforward calculations of the surface SC lead to

    N=(tsinh(2s)t2+2v2,tcosh(2s)t2+2v2,vt22+v2),

    and

    E= t 2+2v2, F=2 t ,G=1,
    e=2 t 2 t 2+2v2, f= t  t 22+v2,g=0.

    The Gaussian and mean curvatures of SC are

    K=2 t 2( t 2+2v2)2,H= t 2( t 22(1+v2))( t 2+2v2)3/2. (4.6)

    In the light of these results, this surface is neither developable nor minimal.

    Likewise, the parametric representation for the ruled surface SW reads as (see Figure 4)

    SW=(tsinh(2s),tcosh(2s),ts+v). (4.7)
    Figure 4.  The evolved W-ruled surface SW of the curve r.

    Making some special calculations related to the considered surface, we get

    N=(sinh(2s),cosh(2s),0),

    and

    E= t 2,F= t ,G=1,e=2 t ,f=0,and g=0.

    The Gaussian and mean curvatures are

    K=0,H= t 2,

    which describe SC as developable and not minimal.

    In the three-dimensional Lorentz–Minkowski 3-space E31, we examined the evolution equations for specific types of ruled surfaces that have significant geometric and physical applications. We employed the alternative frame associated with the basic curve of these surfaces and investigated their key geometric properties. Through a comprehensive analysis, we gained deeper insights into the dynamics of the local curvatures exhibited by these surfaces during their evolution. This work advanced the understanding of the dynamical behavior of surfaces in Lorentz–Minkowski 3-space, with potential implications across various disciplines. Finally, we discussed applications of our preliminary findings, which contribute significantly to the broader field of differential geometry. Looking forward, several avenues for future research emerge. Our findings could be applied to physical models involving particle trajectories or gravitational fields, potentially revealing how timelike ruled surfaces might be utilized in practical scenarios or theoretical frameworks. Further studies could explore alternative frames in more complex settings or for other types of surfaces beyond ruled ones, which might lead to a broader understanding of geometric structures in relativity. The study of solitons through their evolution equations provides crucial insights into a wide range of phenomena, contributing to advances in both theoretical and applied sciences. Solitons are special types of solutions to nonlinear partial differential equations that maintain their shape while propagating at constant velocity. They are often associated with topological solutions and are significant in various fields [16]. Here are some potential applications of evolution equations related to solitons (topological solutions): In the field of mathematical physics, the study of solitons is essential for understanding integrable systems, where the evolution equations are exactly solvable. This has applications in theoretical models and helps in the development of new mathematical techniques. Also, in plasma physics, the evolution equations used in solitons can describe stable wave packets in plasma, which helps in understanding space weather–phenomena and the behavior of high-energy plasmas. By expanding on these aspects, in future research, we will combine the results and methods in [25,26,27] to deepen our knowledge and uncover new applications of differential geometry in both theoretical and practical contexts.

    Yanlin Li: Conceptualization, Methodology; M. Khalifa Saad: Validation, Formal analysis; H. S. Abdel-Aziz: Writing-review, Supervision, Validation; H. M. Serry: Methodology, Writing-original draft; F. M. El-Adawy: Validation, editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors express gratitude to the reviewers and editors for their helpful comments and suggestions.

    All authors declare no conflicts of interest in this paper.



    [1] M. Önder, H. H. Uĝurlu, Frenet frames and invariants of timelike ruled surfaces, Ain Shams Eng. J., 4 (2013), 507–513. https://doi.org/10.1016/j.asej.2012.10.003 doi: 10.1016/j.asej.2012.10.003
    [2] R. López, Differential geometry of curves and surfaces in Lorentz-Minkowski space, Int. Electron. J. Geom., 7 (2014), 44–107.
    [3] Y. Li, E. Güler, M. Toda, Family of right conoid hypersurfaces with light-like axis in Minkowski four-space, AIMS Mathematics, 9 (2024), 18732–18745. https://doi.org/10.3934/math.2024911 doi: 10.3934/math.2024911
    [4] Y. Li, E. Güler, Right conoids demonstrating a time-like axis within Minkowski four-dimensional space, Mathematics, 12 (2024), 2421. https://doi.org/10.3390/math12152421 doi: 10.3390/math12152421
    [5] B. Yilmaz, H. Aykut, Alternative partner curves in the Euclidean 3-space, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 69 (2020), 1–10. https://doi.org/10.31801/cfsuasmas.538177 doi: 10.31801/cfsuasmas.538177
    [6] H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, M. K. Saad, Geometry and evolution of Hasimoto surface in Minkowski 3-space, PLoS One, 19 (2024), e0294310. https://doi.org/10.1371/journal.pone.0294310 doi: 10.1371/journal.pone.0294310
    [7] R. M. Wald, General relativity, University of Chicago Press, 1984.
    [8] S. Manukure, Y. Zhou, A (2+1)-dimensional shallow water equation and its explicit lump solutions, Int. J. Mod. Phys. B, 33 (2019), 1950038. https://doi.org/10.1142/S0217979219500383 doi: 10.1142/S0217979219500383
    [9] D. Yu, F. Cao, Construction and approximation degree for feedforward neural networks with sigmoidal functions, J. Comput. Appl. Math., 453 (2025), 116150. https://doi.org/10.1016/j.cam.2024.116150 doi: 10.1016/j.cam.2024.116150
    [10] Y. Li, M. Aquib, M. A. Khan, I. Al-Dayel, K. Masood, Analyzing the Ricci tensor for slant submanifolds in locally metallic product space forms with a semi-symmetric metric connection, Axioms, 13 (2024), 454. https://doi.org/10.3390/axioms13070454 doi: 10.3390/axioms13070454
    [11] Y. Li, M. Aquib, M. A. Khan, I. Al-Dayel, M. Z. Youssef, Geometric inequalities of slant submanifolds in locally metallic product space forms, Axioms, 13 (2024), 486. https://doi.org/10.3390/axioms13070486 doi: 10.3390/axioms13070486
    [12] A. M. G. Ahmed, A. Adjiri, S. Manukure, Soliton solutions and a bi-Hamiltonian structure of the fifth-order nonlocal reverse-spacetime Sasa-Satsuma-type hierarchy via the Riemann-Hilbert approach, AIMS Mathematics, 9 (2024), 23234–23267. https://doi.org/10.3934/math.20241130 doi: 10.3934/math.20241130
    [13] E. A. Appiah, S. Manukure, An integrable soliton hierarchy associated with the Boiti-Pempinelli-Tu spectral problem, Mod. Phys. Lett. B, 35 (2021), 2150282. https://doi.org/10.1142/S0217984921502821 doi: 10.1142/S0217984921502821
    [14] Y. Li, M. D. Siddiqi, M. A. Khan, I. Al-Dayel, M. Z. Youssef, Solitonic effect on relativistic string cloud spacetime attached with strange quark matter, AIMS Mathematics, 9 (2024), 14487–14503. https://doi.org/10.3934/math.2024704 doi: 10.3934/math.2024704
    [15] Y. Li, A. Gezer, E. Karakas, Exploring conformal soliton structures in tangent bundles with Ricci-Quarter symmetric metric connections, Mathematics, 12 (2024), 2101. https://doi.org/10.3390/math12132101 doi: 10.3390/math12132101
    [16] S. Manukure, T. Booker, A short overview of solitons and applications, Partial Differ. Equ. Appl. Math., 4 (2021), 100140. https://doi.org/10.1016/j.padiff.2021.100140 doi: 10.1016/j.padiff.2021.100140
    [17] S. Manukure, A. Chowdhury, Y. Zhou, Complexiton solutions to the asymmetric Nizhnik-Novikov-Veselov equation, Int. J. Mod. Phys. B, 33 (2019), 1950098. https://doi.org/10.1142/S021797921950098X doi: 10.1142/S021797921950098X
    [18] N. H. Abdel-All, R. A. Hussien, T. Youssef, Evolution of curves via the velocities of the moving frame, J. Math. Comput. Sci., 2 (2012), 1170–1185.
    [19] H. S. Abdel-Aziz, H. M. Serry, F. M. El-Adawy, A. A. Khalil, On admissible curves and their evolution equations in pseudo-galilean space, J. Math. Comput. Sci., 25 (2021), 370–380. https://doi.org/10.22436/jmcs.025.04.07 doi: 10.22436/jmcs.025.04.07
    [20] G. U. Kaymanli, C. Ekici, M. Dede, Directional evolution of the ruled surfaces via the evolution of their directrix using q-frame along a timelike space curve, Avrupa Bilim ve Teknoloji Dergisi, 20 (2020), 392–396. https://doi.org/10.31590/EJOSAT.681674 doi: 10.31590/EJOSAT.681674
    [21] T. Frankel, The Geometry of physics: An introduction, Cambridge University Press, 2011.
    [22] R. López, E. Demir, Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, Open Math., 12 (2014), 1349–1361. https://doi.org/10.2478/s11533-014-0415-0 doi: 10.2478/s11533-014-0415-0
    [23] H. S. Abdel-Aziz, H. Serry, M. K. Saad, Evolution equations of pseudo spherical images for timelike curves in minkowski 3-space, Math. Stat., 10 (2022), 884–893. https://doi.org/10.13189/ms.2022.100420 doi: 10.13189/ms.2022.100420
    [24] H. S. Abdel-Aziz, M. K. Saad, A. A. Abdel-Salam, On involute-evolute curve couple in the hyperbolic and de sitter spaces, J. Egypt. Math. Soc., 27 (2019), 25. https://doi.org/10.1186/s42787-019-0023-z doi: 10.1186/s42787-019-0023-z
    [25] B. Ó Neill, Semi-Riemannian geometry with applications to relativity, Academic press, 1983.
    [26] B. Sahiner, Ruled surfaces according to alternative moving frame, 2019, arXiv: 1910.06589. https://doi.org/10.48550/arXiv.1910.06589
    [27] Y. Zhou, S. Manukure, C. Zhang, X. Zhang, Resonant solutions and breathers to the BKP equation, Nonlinear Dyn., 111 (2023), 8611–8616. https://doi.org/10.1007/s11071-023-08253-9 doi: 10.1007/s11071-023-08253-9
  • This article has been cited by:

    1. Ştefan-Cezar Broscăţeanu, Adela Mihai, Andreea Olteanu, A Note on the Infinitesimal Bending of a Rectifying Curve, 2024, 16, 2073-8994, 1361, 10.3390/sym16101361
    2. Yanlin Li, Arup Kumar Mallick, Arindam Bhattacharyya, Mića S. Stanković, A Conformal η-Ricci Soliton on a Four-Dimensional Lorentzian Para-Sasakian Manifold, 2024, 13, 2075-1680, 753, 10.3390/axioms13110753
    3. Yanlin Li, Nasser Bin Turki, Sharief Deshmukh, Olga Belova, Euclidean hypersurfaces isometric to spheres, 2024, 9, 2473-6988, 28306, 10.3934/math.20241373
    4. Guangyuan Tian, Xianji Meng, Exact Solutions to Fractional Schrödinger–Hirota Equation Using Auxiliary Equation Method, 2024, 13, 2075-1680, 663, 10.3390/axioms13100663
    5. Yanlin Li, M. S. Siddesha, H. Aruna Kumara, M. M. Praveena, Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds, 2024, 12, 2227-7390, 3130, 10.3390/math12193130
    6. Yanlin Li, Kemal Eren, Soley Ersoy, Ana Savić, Modified Sweeping Surfaces in Euclidean 3-Space, 2024, 13, 2075-1680, 800, 10.3390/axioms13110800
    7. Fawaz Alharbi, Yanlin Li, Vector fields on bifurcation diagrams of quasi singularities, 2024, 9, 2473-6988, 36047, 10.3934/math.20241710
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(879) PDF downloads(77) Cited by(7)

Figures and Tables

Figures(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog