Abstract
In this study, we consider timelike revolution hypersurfaces of constant ratio in Minkowski space-time. At first, we exhibit the representations of revolution hypersurfaces given by three different forms. Then, 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 .
1. Introduction
The concept of constant ratio submanifolds was first discussed by Chen in 2001, and then many researchers evaluated this concept on curves and surfaces from different perspectives [1–6].
Let be a hypersurface in Minkowski 4–space. The parameterization of the hypersurface can be separated into tangent component and normal component as
The name “constant ratio” comes from the ratio of this tangent component and the normal component. Denoting the orthonormal frame and the distance function , the gradient of is known as
Moreover, by the use ofequation (2) becomes
Here, indicates the Lorentzian metric in . 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 flat (developable surface) [7–9].
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. Then, we yield the conditions for them to become constant ratio surface. We classify these types of hypersurfaces with respect to satisfying and [1, 2].
2. Preliminaries
In Minkowski space-time, the Lorentzian metric is given byand the vector product is known aswhere , , and .
A vector in is called as timelike, null, or spacelike with respect to satisfying , , or , respectively. Also, the norm of this vector is presented by
In Minkowski space-time, a hypersurfaceis named as timelike (spacelike), based on its unit normal vector (or Gauss map) being spacelike (timelike), and the normal vector field is calculated by
The matrix that corresponds to the first fundamental form is [10]where the coefficients are
For a timelike hypersurface, the coefficient E, G, or C is negative definite.
In four-dimensional Minkowski space, with the help of spacelike, timelike, and lightlike axis spanned by (0, 0, 0, 1), (1, 0, 0, 0), and (1, 1, 0, 0), the three rotation matrices are given by [11]
3. Hypersurfaces of Constant Ratio in Four-Dimensional Minkowski Space
Definition 1. Let be a hypersurface in Minkowski space-time. In case of the norm of being positive real constant, is said to be constant ratio surface:As it can be understood from the definition, satisfying the condition means thatBy the use of (16) and the inequality we can say
Let be the orthonormal frame in can be considered as parallel to Therefore, the following relations can be written:where and are differentiable functions.
In case the hypersurface is of constant ratio, we getTherefore,
3.1. Timelike Revolution Hypersurfaces of Type I
Definition 2. Let be a smooth function and be a curve on a plane parameterized by in The surface formed by the rotation of the curve around the spacelike axis is called as revolution hypersurface of type I. Therefore, with the help of the matrix the parameterization of is given byThe tangent vector fields areUsing vector product (11), the unit normal vector is calculated bywhere . Since we suppose the surface is timelike, Let the first unit tangent vector be parallel to and timelike ( is negative definite). Then, by the use of (9) and (21), we writeand denote With the help of the relationwe getThus, the functions and are
3.2. Timelike Revolution Hypersurfaces of Type II
Definition 3. Let be a smooth function and be a curve on a plane parameterized by in The surface formed by the rotation of the curve around the timelike axis is called as revolution hypersurface of type II. Therefore, with the help of the matrix the parameterization of is given byThe tangent vector fields areUsing vector product (14), the unit normal vector is calculated bywhere . Since we suppose the hypersurface is timelike, the unit normal vector field is spacelike (). Let the first unit tangent vector be parallel to and timelike. Using and (9), we writeand denote With the help of the relationwe getThus, the functions and are
3.3. Timelike Revolution Hypersurfaces of Type III
Definition 4. Let be a smooth function and be a curve on a plane parameterized by in The surface formed by the rotation of the curve around the lightlike axis is called as revolution hypersurface of type III. Therefore, with the help of the matrix the parameterization of is given bywhere
This parameterization can be written asThe tangent vector fields areUsing vector product (11), the unit normal vector is calculated bywhere . Since we suppose the surface is timelike, Let the first unit tangent vector be parallel to and timelike. Then, by the use of and (9), we noteand denote With the help of the relationwe get the functions and as
3.4. Results for Timelike Revolution Hypersurfaces of Constant Ratio in
Theorem 5. Let be a hypersurface of revolution given by (20), (29), or (37). Then, corresponds to a constant ratio surface satisfying if and only if the differentiable function is presented bywhere for (20) and (37), for (29).
Proof. Let be a constant ratio hypersurface of revolution with Using (6) and (19), In this case, must be constant (see [1]). Thus, one can writeBy the use of (28) or (35) or (42), we obtain the differential equationswhich have the solutionThis completes the proof.
Theorem 6. Let be a hypersurface of revolution given by (20), (29), or (37). Then, corresponds to a constant ratio surface satisfying if and only if the differentiable function is presented bywhere is a real constant.
Proof. Let be a constant ratio hypersurface of revolution with Using (6) and (19), andBy the use of (28) or (35) or (42), we obtain the differential equationswhich have the solutionwhere This completes the proof.
Theorem 7. Let be a hypersurface of revolution given by (20), (29), or (37). Then, corresponds to a constant ratio surface satisfying , , if and only ifholds.
Proof. Let be a constant ratio hypersurface of revolution with , Then, for the length of the position vector of ,is satisfied (see, [1, 2]). With the help of this equation and (16), we getCombining (28) or (35) or (42) with (43) and (54), we getwhich indicates (51). This completes the proof.
Example 8. Taking and in the parameterization (20), we can plot the projection of the hypersurface of constant ratio shown in Figure 1 by using Maple command:
4. Conclusion
Constant ratio submanifolds are among the significant classifications in differential geometry. In this work, constant ratio hypersurfaces in Minkowski space-time are discussed on the parameterizations of revolution hypersurfaces according to three rotations. Some different characterizations of these types of hypersurfaces can be investigated in future studies.
Data Availability
No data were used to support the findings of this article.
Conflicts of Interest
The author declares that there are no conflicts of interest.