A New Approach to Timelike Hypersurfaces of Constant Ratio in IE 41

In this study, we consider timelike revolution hypersurfaces of constant ratio in Minkowski space-time. At frst, we exhibit the representations of revolution hypersurfaces given by three diferent forms. Ten, we yield the conditions for such hypersurfaces to correspond to constant ratio surface. As a result of these conditions, we present the position vectors of constant ratio timelike rotational hypersurfaces in IE 41 .

Let S: ψ(u, v, w): (u, v, w) ∈ D(D ⊂ E 3 ) be a hypersurface in Minkowski 4-space.Te parameterization of the hypersurface can be separated into tangent component and normal component as Te name "constant ratio" comes from the ratio of this tangent component and the normal component.Denoting the orthonormal frame σ 1 , σ 2 , σ 3 , σ 4   and the distance function ρ � ‖ψ‖, the gradient of ρ is known as grad(ρ) �  ( Moreover, by the use of Here, g indicates the Lorentzian metric in IE 4  1 .Hence, the norm of gradient function is congruent to the equality ( If this relation is equal to a positive constant, then related surface is called as constant ratio surface, i.e., Revolution surface that has many applications in multidisciplinary sciences are also used theoretically in geometry with the forms catenoid, tube surface, canal surface, ruled, and developable surface.Some of them have characteristic features as being minimal (catenoid) and being fat (developable surface) [7][8][9].
In the present work, we evaluate the timelike constant ratio hypersurfaces of revolution in four-dimensional Minkowski space.Firstly, we present the three types of parameterizations of rotational hypersurfaces.Ten, we yield the conditions for them to become constant ratio surface.We classify these types of hypersurfaces with respect to satisfying ‖grad(ρ)‖ � 0,‖grad(ρ)‖ � 1, and ‖grad(ρ

Preliminaries
In Minkowski space-time, the Lorentzian metric is given by and the vector product is known as ), and w � (w 1 , w 2 , w 3 , w 4 ).
A vector u in IE 4  1 is called as timelike, null, or spacelike with respect to satisfying g(u, u) < 0, g(u, u) � 0, or g(u, u) > 0, respectively.Also, the norm of this vector is presented by In Minkowski space-time, a hypersurface is named as timelike (spacelike), based on its unit normal vector (or Gauss map) being spacelike (timelike), and the normal vector feld is calculated by Te matrix that corresponds to the frst fundamental form is [10] where the coefcients are For a timelike hypersurface, the coefcient E, G, or C is negative defnite.

Hypersurfaces of Constant Ratio in Four-Dimensional Minkowski Space
Defnition 1.Let S: ψ(u, v, w): (u, v, w) ∈ D(D ⊂ E 3 ) be a hypersurface in Minkowski space-time.In case of the norm of grad(ρ) being positive real constant, S is said to be constant ratio surface: As it can be understood from the defnition, satisfying the condition ‖grad(ρ)‖ � k means that By the use of ( 16) and the inequality ‖ψ T ‖ ≤ ‖ψ‖, we can say k ≤ 1.
Let σ 1 , σ 2 , σ 3 , σ 4   be the orthonormal frame in IE 4  1 .σ 1 can be considered as parallel to ψ T .Terefore, the following relations can be written: where a and b are diferentiable functions.
In case the hypersurface is of constant ratio, we get Terefore, (19)

Timelike Revolution Hypersurfaces of Type I
Defnition 2. Let r be a smooth function and C: I ⊂ IR → π be a curve on a plane parameterized by C(u) � (u, 0, 0, r(u)) in IE 4  1 .Te surface S formed by the rotation of the curve C around the spacelike axis (0, 0, 0, 1) is called as revolution hypersurface of type I. Terefore, with the help of the matrix R 1 , the parameterization of S is given by (20) Te tangent vector felds are Using vector product (11), the unit normal vector is calculated by where r � r(u).Since we suppose the surface is timelike, 1 − (r ′ ) 2 > 0. Let the frst unit tangent vector σ 1 be parallel to ψ T and timelike ((σ 1 , σ 1 ) is negative defnite).Ten, by the use of ( 9) and (21), we write and denote σ 4 � N.With the help of the relation we get Tus, the functions a and b are . (28)

Timelike Revolution Hypersurfaces of Type II
Defnition 3. Let r be a smooth function and C: I ⊂ IR → π be a curve on a plane parameterized by C(u) � (r(u), 0, 0, u) in IE 4  1 .Te surface S formed by the rotation of the curve C around the timelike axis (1, 0, 0, 0) is called as revolution hypersurface of type II.Terefore, with the help of the matrix R 2 , the parameterization of S is given by

(30)
Using vector product (14), the unit normal vector is calculated by where r � r(u).Since we suppose the hypersurface is timelike, the unit normal vector feld is spacelike ((r ′ ) 2 − 1 > 0).Let the frst unit tangent vector σ 1 be parallel to ψ T and timelike.Using ψ u and ( 9), we write and denote σ 4 � N.With the help of the relation we get Journal of Mathematics Tus, the functions a and b are (35)

Timelike Revolution Hypersurfaces of Type III
Defnition 4. Let r be a smooth function and C: I ⊂ IR → π be a curve on a plane parameterized by C(u) � (u, r(u), 0, 0) in IE 4  1 .Te surface S formed by the rotation of the curve C around the lightlike axis (1, 1, 0, 0) is called as revolution hypersurface of type III.Terefore, with the help of the matrix R 3 , the parameterization of S is given by where u ∈ R − 0 { }.Tis parameterization can be written as Te tangent vector felds are Using vector product (11), the unit normal vector is calculated by where r � r(u).Since we suppose the surface is timelike, 1 − (r ′ ) 2 > 0. Let the frst unit tangent vector σ 1 be parallel to ψ T and timelike.Ten, by the use of ψ u and (9), we note we get the functions a and b as (42)

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Theorem .Let S be a hypersurface of revolution given by ( 20), ( 29), or (37).Ten, S corresponds to a constant ratio surface satisfying ‖grad(d)‖ � 1 if and only if the diferentiable function r(u) is presented by where c 1 is a real constant.
Proof.Let S: ψ(u, v, w) be a constant ratio hypersurface of revolution with ‖grad(d)‖ � 1.Using ( 6) and (19), k � 1 and By the use of (28) or ( 35) or (42), we obtain the differential equations which have the solution where c 1 ∈ R. Tis completes the proof.

Conclusion
Constant ratio submanifolds are among the signifcant classifcations in diferential geometry.In this work, constant ratio hypersurfaces in Minkowski space-time are discussed on the parameterizations of revolution hypersurfaces according to three rotations.Some diferent characterizations of these types of hypersurfaces can be investigated in future studies.

4
Journal of Mathematics and denote σ 4 � N.With the help of the relation