Timelike Tangent Developable Surfaces and Bonnet Surfaces
A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.
Surfaces which admit a one-parameter family of isometric deformations preserving the mean curvature are called Bonnet surfaces. In 1867, Bonnet proved that any surface with constant mean curvature in (which is not totally umbilical) is a Bonnet surface . Cartan obtained some detailed results for Bonnet surfaces in . Lawson extended Bonnet’s results to any surface with constant mean curvature in Riemannian 3-manifold of constant curvature. Also, it was proved that any Bonnet surface of nonconstant mean curvature depends on six arbitrary constants . Characterization for isometric deformation preserving the principal curvatures of surfaces was obtained by the aid of differential forms by Chern in . The geometric characterizations of helicoidal surfaces of constant mean curvature, helicoidal surfaces as Bonnet surfaces, and tangent developable surfaces as Bonnet surfaces were studied by Roussos in ,  and , respectively. Roussos obtained a characterization for isometric deformation preserving the mean curvature by using the method of Chern. Soyuçok gave the necessary and sufficient condition of a surface to be Bonnet surface, which is to have a special system of isothermal parameters . Moreover, Soyuçok proved that 3-dimensional hyperspace in 4-dimensional space is Bonnet surface if and only if hypersurface has orthogonal net . Bağdatlı and Soyuçok studied hypersurfaces preserving the mean curvature and proved that a hypersurface in is Bonnet surface if and only if hypersurface has orthogonal A-net . On the other hand, Chen and Li studied 3-dimensional Minkowski space and classified timelike Bonnet surfaces .
Let be a timelike surface in 3-dimensional Minkowski space with nondegenerate metric tensor , where is a system of the canonical coordinates in . Let be a timelike immersion that admits a nontrivial isometry preserving the mean curvature. Nontriviality means that the immersion in the family is not in the form of , where is an immersion of . These kinds of surfaces are called timelike Bonnet surfaces by Chen and Li in . Suppose that is a local orthonormal frame at the point , where is a timelike tangent vector, is a spacelike unit tangent vector, and is a spacelike unit normal vector field at . can be regarded as a map , where is the de Sitter space. Let , , be dual 1-forms of defined by , , and let , , be connection forms; then such that , , and .
The Weingarten map is given by and has real eigenvector if and only if ; that is, . Here, and are the mean and Gaussian curvatures of , respectively.
Unless otherwise stated throughout this paper, we will assume that and are the eigenvectors. Thus, and where and are the principal curvatures throughout and . Then, the mean and Gaussian curvatures of the surface are respectively, and . Suppose that ; then and can be determined from the Cartan structure equations given by The Gaussian and Codazzi equations are respectively . If we substitute the equations of (3) into the Codazzi equations, respectively, and make them equal to the exterior differentiations of (3), then we give After reformulating the equations of (8), we can give Since , these last two equations give us Here, if the functions and are defined as and , then the last equation becomes that is, . Moreover, from (9), we get Thus, by the aid of these last two equations, we see If we take into consideration (11), then the gradient of the mean curvature function is Thus, it is easily seen that Here, we assume that is nonnull vector field; that is, . By taking , we find By the fact that Hodge operator is defined as the connection form given by (3) becomes Let us define Then, the following equations are obvious: By considering the second equation of (21), if we rearrange (11), we get In a similar way, from (18) and (20), (13) becomes Thus, we find
3. Timelike Bonnet Surfaces in Minkowski Space
Since we will make use of it in the following section, let us briefly recall construction of the criterion given by  for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space.
Let be another timelike surface in with the principal direction vectors such that it is an isometric deformation of preserving the first fundamental form and principal curvatures. Suppose that is a principal coframe corresponding to orthonormal of ; then the first fundamental form of is and the principal curvatures throughout and are respectively. It is seen from (26) that there is a function on such that By direct calculations and the exterior derivative of (28), we get and . From the first Cartan structure equations, we obtain On the other hand, from (25), we write . Equations (27) and (29) and the last equation give us . If we apply operator to this equality, we find . By the fact that , , we see . After the necessary arrangements, we find . If we consider (29), we get is satisfied for with respect to (23). Considering (27) and comparing the last equation with (23), we see . Thus, and if we consider (28) we write By taking (20), if we substitute (28) and (31) into , we obtain Let us define . The differentiation of is The total differential equation (33) is satisfied by rotations of the principal directions with hyperbolic angles during isometric deformations. The deformation is nontrivial if and only if (33) is completely integrable .
Theorem 1. Every timelike constant mean curvature surface with in has one-parameter family of nontrivial isometric deformations preserving the mean curvature; that is, is timelike Bonnet surface .
In order to investigate the situations of being constant or nonconstant, let thus defining and . By substituting (34) into the exterior derivative of (33), we get Thus, the following classification is satisfied: constant. constant, , and . constant, , and .
If these categories are investigated separately, then , and can be interpreted as follows.
Since the mean curvature is nonconstant and and in , (35) is satisfied for all .
The mean curvature is nonconstant and and in . In this case, by considering (35), we get can be computed for any timelike surface with nonconstant mean curvature, nonnull and but in order to find the nontrivial isometry preserving the mean curvature given in (36) must satisfy (33). Thus, constitutes a criterion for being timelike Bonnet surface.
In the following section, we will check this criterion for timelike tangent developable surfaces and obtain the necessary and sufficient condition for timelike tangent developable surfaces to belong to case .
4. Timelike Tangent Developable Surfaces
Timelike tangent developable surfaces can be investigated in two subcases when the curvature and torsion of base curve of these surfaces are nonconstant and constant.
4.1. Timelike Tangent Developable Surfaces with Base Curve with Nonconstant Curvature and Torsion
Let be a timelike tangent developable surface given by where is timelike curve parametrized by its arc-length and is timelike unit tangent vector field. Also assume that . (In the same way, gives the second sheet of this surface.) Let be principal normal vector field of and of course it is spacelike. Then, such that is the curvature of the base curve . From (38), it is seen that and . The first fundamental form of this surface is Then, can be given from . Also, keeping in mind , we see By the fact that , we get On the other hand, since , that is, , we find . By defining , is binormal vector field of base curve and spacelike. Let us denote the torsion of by . Considering the Serret-Frenet formulae, we obtainMoreover, we get since . From the fact that is the dual coframe corresponding to principal frame field , the principal curvatures throughout the principal direction vector fields and satisfy . That is, since , it is assumed that . By taking the principal curvatures in terms of curvature and torsion of base curve, and can be given as respectively. On the other hand, from (11), we get If we substitute (40) into this last equation, we find If we call by considering (42) from (44) we obtain If we substitute (40) and (46) into (20), we get The exterior differentiations of the equalities given in (47) arerespectively. The exterior product of the equalities given in (47) is such that it never vanishes. By putting (48) and (49) into the equalities in (34), we find In order to check whether the criterion of being a timelike Bonnet surface given in the case of is satisfied or not, let us substitute the equations given in (50) into (36) and find By arranging (45), we get Thus, (51) becomesFrom exterior derivative of (53), we obtain In order to find the right side of (33), we consider (47) and (53) and find If (54) and (55) are compared, is obtained since . If we write this equation into (52), we get If we substitute (56) into (52) and solve it, we find where is a constant value such that . If (56) and (57) are written in (54), is obtained. By comparing (55) with (59) and substituting into (56), we get It is easily seen thatthat is,where is constant. From this last equation, we get If we put (56) and (58) into (62), then we find By multiplying each side of the last equation by , we get Here, assume that . Thus, (68) becomes The solution of this differential equation is Here, and . Thus, there is the inequality If , that is, , then or and , where and are the roots of the quadratic trinomial .
Thus, the following theorem can be given.
Theorem 2. Timelike tangent developable surface is a timelike Bonnet surface if and only if
4.2. Timelike Tangent Developable Surfaces with Base Curve with Constant Curvature and Torsion
If the curvature and torsion of timelike base curve of timelike tangent developable surface are constants, then the base curve is timelike circular helix. All -constant curves are helices and is a timelike flat helicoidal surface. Since , , and are determined as in Section 4.1, we have By considering the possibility of reflection, the forms of the image surface are Moreover, the image surface is flat but it is not a cylinder or a cone because of ; that is, is satisfied for cylinders and cones. Thus, a timelike developable surface cannot be a cylinder or a cone. Therefore, image surface is timelike developable surface given by , . Here, is the arc-length of the timelike base curve and is timelike unit tangent vector. The forms of timelike surfaces are
By comparing the forms given in (70) with (71), one can see that should be taken with the sign + whereas should be taken with the sign −; that is, the reflection is necessary and , , , . Thus, the image surface is the original surface and the nontrivial isometry preserving the mean curvature is or Thus, the following corollaries can be given.
Corollary 3. The mapping is a timelike isometry preserving the mean curvature. The second fundamental forms of corresponding surfaces are different; that is, is nontrivial.
Corollary 4. , is also timelike nontrivial isometry from to its other sheet preserving mean curvature.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
O. Bonnet, “Mémoire sur la théorie des surfaces applicables,” Journal de l'École Polytechnique, vol. 42, pp. 72–92, 1867.View at: Google Scholar
E. Cartan, “Sur les couples de surfaces applicables avec conservation des courbures principales,” Bulletin des Sciences Mathématiques, vol. 66, no. 2, pp. 55–72, 74–85, 1942.View at: Google Scholar
H. B. Lawson, “Complete minimal surface in S3,” Annals of Mathematics, vol. 92, no. 2, pp. 335–374, 1970.View at: Google Scholar
Z. Soyuçok, “The problem of isometric deformations of a Euclidean hypersurface pre-serving mean curvature,” Bulletin of the Technical University of Istanbul, vol. 49, no. 3-4, pp. 551–562, 1996.View at: Google Scholar
W. Chen and H. Li, “On the classification of the timelike Bonnet surfaces,” in Geometry and Topology of Submanifolds 10, Differential Geometry in Honor of Professor S S Chern, pp. 18–31, Peking University, Beijing, China; TU Berlin, Berlin, Germany, 1999.View at: Google Scholar