Abstract
We consider hyperbolic rotation (), hyperbolic translation (), and horocyclic rotation () groups in , which is called Minkowski model of hyperbolic space. Then, we investigate extrinsic differential geometry of invariant surfaces under subgroups of in . Also, we give explicit parametrization of these invariant surfaces with respect to constant hyperbolic curvature of profile curves. Finally, we obtain some corollaries for flat and minimal invariant surfaces which are associated with de Sitter and hyperbolic shape operator in .
1. Introduction
Hyperbolic space has five analytic models, which are isometrically equivalent to each other [1, 2]. In this study, we choose Minkowski model of hyperbolic space which is denoted by . In a different point of view, we may consider invariant surface as rotational surface. In this sense, rotational surfaces in different ambient spaces were studied by many authors. For instance, in [3], Carmo and Dajczer define rotational hypersurfaces with constant mean curvature (cmc) in hyperbolic -space. They also give a local parametrization of this surface in terms of the cmc under some special conditions. In [4], Mori studied elliptic, spherical, and parabolic type rotational surfaces with cmc in . In [5], the total classification of the timelike and spacelike hyperbolic rotation surfaces is given in terms of cmc in 3-dimensional de Sitter space . As a general form, explicit parametrizations of rotational surfaces with cmc are given in Minkowski -space by [6].
This paper is organized as follows. In Section 2, we give briefly the notions of H-point, H-line, H-plane, and H-distance in hyperbolic geometry of . Throughout this work, the prefix “H-” is used is the sense of belonging to hyperbolic space. It is well known that H-isometry is a map which is preserved H-distance in . The set of H-isometries is a group which is identified with restriction of isometries of Minkowski 4-space to . Let the group of H-isometries be denoted by in . We consider subgroups , , and of with respect to leaving fixed timelike, spacelike, and lightlike planes of , respectively. Then, we give the notions of H-rotation, H-translation, and horocyclic rotation which are one-parameter actions of in . Moreover, we obtain some properties of H-isometries. There exist three kinds of totally umbilical surfaces which are called H-sphere, equidistant surface, and horosphere in . We obtain a classification of H-isometries by the subgroups , , and with respect to leaving fixed equidistant surfaces, H-spheres, and horospheres in , respectively. In Section 3, we give the basic theory of extrinsic differential geometry of curves and surfaces in . In Section 4, we investigate surfaces which are invariant under a subgroup of H-translations in . Moreover, in the sense of de Sitter and hyperbolic shape operator in , we study extrinsic differential geometry of these invariant surfaces by using notations in [7, 8]. We give a relation between one of the principal curvatures of the invariant surface and hyperbolic curvature of profile curve of the invariant surface in . In a different viewpoint, we obtain explicit parametrization of some invariant surfaces in terms of constant hyperbolic curvature of profile curve. Moreover, we give some geometric results with respect to constant hyperbolic curvature of profile curve for flat and minimal invariant surfaces in . Finally, we give a classification theorem for the totally umbilical invariant surfaces in .
2. Isometries of
In [9], Reynold give a brief introduction to hyperbolic geometry of hyperbolic plane . Also, he described explicit descriptions of the hyperbolic metric and the isometries of the hyperbolic plane. In this section, we consider hyperbolic geometry in . We especially determine isometry groups of with respect to causal character of hyperplanes of ; then, these isometry groups are classified in terms of leaving those totally umbilic surfaces of fixed.
Let denote the 4-dimensional Minkowski space, that is, the real vector space endowed with the scalar product for all , . Let be pseudo-orthonormal basis for . Then, for signatures , . The function is called the associated quadratic form of .
A vector is called spacelike, timelike, and lightlike if (or ), , and , respectively. The Lorentzian norm of a vector is defined by .
The sets are called Minkowski model of hyperbolic space, de Sitter space, and future light cone, respectively.
Let be a vector subspace of . Then, is said to be timelike, spacelike, and lightlike if and only if contains a timelike vector and every nonzero vector in is spacelike otherwise, respectively.
Now, we give basic notions for hyperbolic geometry in . From now on, we use the prefix “H-” instead of “hyperbolic” for brevity.
An H-point is intersection such that is 1-dimensional timelike subspace of and is called . An H-line is intersection such that is 2-dimensional timelike subspace of and is called . An H-plane is intersection such that is 3-dimensional timelike subspace of and is called .
H-coordinate axes are denoted by intersections such that for . H-coordinate planes are denoted by intersections such that for . H-upper (H-lower) half-spaces of are defined by intersections and upper (lower) half-space of .
A hyperplane in is defined by for a pseudo-normal and a real number . If is spacelike, timelike, or lightlike, is called timelike, spacelike, or lightlike, respectively.
Three kinds of totally umbilic surfaces have which are given by intersections of and hyperplanes in . A surface is called H-sphere, equidistant surface, and horosphere if is spacelike, timelike, and lightlike, respectively.
We now give the existence and uniqueness of any H-line or H-plane in . Any given two distinct points determine unique 2-plane through origin in and three distinct points determine unique 3-plane through origin in . So the following propositions are clear.
Proposition 1. Any given two distinct H-points lie on a unique H-line in .
Proposition 2. Any given three distinct H-points lie on a unique H-plane in .
Also, we say that H-line segments , H-ray are determined by two different H-points and in natural way.
Definition 3. Let be parametrization of . Then, H-length of is given by
If we take any hyperbolic space curve instead of in Definition 3, then H-arc length of any hyperbolic space curve is calculated by formula (4) in the same way. Moreover, H-distance between H-points and is given by
Let be a linear transformation. Then is called linear isometry (with respect to ) if it satisfies the following equation: Let matrix form of linear transformation be denoted by , with respect to pseudo-orthonormal basis . The set of all linear isometries of is a group under matrix multiplication and it is denoted by where signature matrix . It is also called semiorthogonal group of and so
The subgroup is called special semiorthogonal group. Let block matrix form of be . Then, and are called timelike and spacelike part of , respectively.
Definition 4. (i) If , then preserves (reverses) time orientation.
(ii) If , then preserves (reverses) space orientation.
Thus, is decomposed into four disjoint sets indexed by the signs of and in that order. They are called , and . We define the group Elements of preserve H-distance in . It is clear that for every and . Thus, we are ready to give the following definition.
Definition 5. Every element of is an H-isometry in .
Thus, we say that is union of subgroup which preserves time orientation of . That is, .
We consider , , and subgroups of which leave fixed timelike, spacelike, and lightlike planes of , respectively. Let matrix representation of H-isometries be , and let H-isometries , , and be denoted by , , and , respectively.
We suppose that . Then, leaves fixed timelike planes for of . So that .
If for , then entries of matrix must be , , , and and , , , and . By using (7) and (8), we have , , , and and the following equation system: If the above system is solved under time orientation preserving and sign cases, then general form of H-isometries that leave fixed timelike plane of is given by such that for all . In other cases, if and , then general forms of H-isometries that leave fixed timelike planes and of are given by such that for all , respectively. Thus, we say that the group is union of disjoint subgroups of and such that that is, .
We suppose that . Then, leaves fixed spacelike planes for of . That is, .
If for , , then entries of must be , , , and and , , , and . By using (7) and (8), we have , , , and and the following equation system: If the above system is solved under time orientation preserving and sign cases, then general form of H-isometries that leave fixed spacelike plane of is given by such that for all . In other cases, if and , then general forms of H-isometries that leave fixed spacelike planes and of are given by such that for all , respectively. Thus, we say that the group is union of disjoint subgroups of and such that that is, .
We suppose that . Then, leaves fixed lightlike planes for of . So that .
If for , , then entries of matrix must be By using (7) and (8), we obtain the following equation system: If the above system is solved under time orientation preserving and sign cases, then general form of H-isometries that leaves fixed lightlike plane of is given by such that for all . In other cases, we apply similar method. Hence, if H-isometry that leaves fixed lightlike plane of is denoted by , then we have the following H-isometries: and we obtained the following general forms: So, we say that the group is union of disjoint subgroups of and such that that is, .
Hence, it is clear that However, we see that easily from matrix multiplication for any .
Let parametrization of H-coordinate axes , , and be and let their matrix forms be respectively. After applying suitable H-isometry to H-coordinate axis for , we obtain the following different parametrizations: for hyperbolic polar coordinates , and . Thus, we see roles of H-isometries and in from equations Moreover, and leave fixed ; that is, Thus, we are ready to give the following definitions by (33) and (38).
Definition 6. is H-translation by along H-coordinate axis for in .
Definition 7. is H-rotation by about H-coordinate axis for in .
Definition 8. is horocyclic rotation by about lightlike plane for () in .
Now, we give corollaries about some properties of H-isometries and transition relation between H-coordinate axes with H-coordinate planes .
Corollary 9. Any H-coordinate axis is converted to each other by suitable H-rotation. That is,
Corollary 10. A H-plane consists in suitable H-coordinate axis and H-rotation. Namely,
Corollary 11. Any H-coordinate plane is converted to each other by suitable H-rotation. That is,
Corollary 12. Any horocyclic rotation is converted to each other by suitable H-rotations. Namely,
After the notion of congruent in , we will give a different classification theorem of H-isometries in terms of leaving those totally umbilic surfaces of fixed.
Definition 13. Let and be two subsets of . If for some , then and are called congruent in .
Theorem 14. An H-sphere is invariant under H-translation in .
Proof. Suppose that is an H-sphere. Then, there exists a spacelike hyperplane with timelike normal such that for . So, Moreover, for and , we have Since , for any hyperbolic polar coordinates , and . If we apply H-isometry , then we have unit timelike vector such that However, unit timelike normal vector is invariant under . That is, For this reason, if is an H-sphere which is generated from spacelike hyperplane then we have by (47). Therefore, is invariant under H-translations. Finally, The proof is completed since and are congruent by (46).
The following theorems also can be proved using similar method.
Theorem 15. An equidistant surface is invariant under H-rotation in .
Theorem 16. A horosphere is invariant under horocyclic rotation in .
Finally, we give the following corollary.
Corollary 17. Equidistant surfaces, H-spheres, and horospheres are invariant under groups , , and in , respectively.
3. Differential Geometry of Curves and Surfaces in
In this section, we give the basic theory of extrinsic differential geometry of curves and surfaces in . Unless otherwise stated, we use the notation in [7, 8].
The Lorentzian vector product of vectors is given by where is the canonical basis of and . Also, it is clear that for any . Therefore, is pseudo-orthogonal to any .
We recall the basic theory of curves in . Let be a unit speed regular curve for open subset . Since , tangent vector of is given by . The vector is orthogonal to and . We suppose that . Then, the normal vector of is given by . However, the binormal vector of is given by . Hence, we have a pseudo-orthonormal frame field of along and the following Frenet-Serret formulas: where hyperbolic curvature and hyperbolic torsion of are given by and under the assumption , respectively.
Remark 18. The condition is equivalent to . Moreover, we see easily that if and only if there exists a lightlike vector such that a geodesic (H-line).
If and , then in is called a horocycle. We give a lemma about existence and uniqueness for horocycles (cf. [8, Proposition 4.3]).
Lemma 19. For any and such that , the unique horocycle with the initial conditions , , and is given by
Now, we recall the basic theory of surfaces in . Let be embedding such that open subset . We denote that regular surface and identify and through the embedding , where is a local chart. For and , if we define spacelike unit normal vector where , then we have , . We also regard as unit normal vector field along in . Moreover, is a lightlike vector since , . Then the following maps , and , are called de Sitter Gauss map and light cone Gauss map of , respectively [8]. Under the identification of and via the embedding , the derivative can be identified with identity mapping on the tangent space at . We have that .
For any given , the linear transforms and are called de Sitter shape operator and hyperbolic shape operator of , respectively. The eigenvalues of and are denoted by and for , respectively. Obviously, and have same eigenvectors. Also, the eigenvalues satisfy where and are called de Sitter principal curvature and hyperbolic principal curvature of at , respectively.
The de Sitter Gauss curvature and the de Sitter mean curvature of are given by at , respectively. Similarly, The hyperbolic Gauss curvature and the hyperbolic mean curvature of are given by at , respectively. Evidently, we have the following relations:
We say that a point is an umbilical point if . Also, is totally umbilical if all points on are umbilical. Now, we give the following classification theorem of totally umbilical surfaces in (cf. [8, Proposition 2.1]).
Lemma 20. Suppose that is totally umbilical. Then, is a constant . Under this condition, one has the following classification. (1)Supposing that , if and , then is a part of an equidistant surface; if and , then is a part of a sphere; if , then is a part of a plane (H-plane).(2)If , then is a part of horosphere.
4. -Invariant Surfaces in
In this section, we investigate surfaces which are invariant under some one parameter subgroup of H-translations in . Moreover, we study extrinsic differential geometry of these invariant surfaces.
Let be a regular surface via embedding such that open subset . We denote by the shape operator of with respect to unit normal vector field in . Let us represent by , and the Levi-Civita connections of , and , respectively. Then the Gauss and Weingarten explicit formulas for in are given by for all tangent vector fields , respectively.
Let be an open interval of and let be a unit speed regular curve which is lying on H-plane . Without loss of generality, we consider the subgroup which is H-translation group along H-coordinate axis in . Let be a regular surface which is given by the embedding where is open subset of . Since for all , is invariant under H-translation group . Hence we say that is a -invariant surface and is the profile curve of in .
Remark 21. From now on, we will not use the parameter “t” in case of necessity for brevity.
By (61) and (62), the parametrization of is for all . By (59), where , , , and . So, we have Hence, is orthogonal tangent frame of . If , then we have that and also for all since . If the unit normal vector of in is denoted by , then we have that and it is clear that for all . From (59) and (60), the matrix of de Sitter shape operator of with respect to orthogonal tangent frame of is at any , where After basic calculations, the de Sitter principal curvatures of are
Let Frenet-Serret apparatus of be denoted by in .
Proposition 22. The binormal vector of the profile curve of -invariant surface is constant in .
Proof. Let be the profile curve of . By (61), we know that is a hyperbolic plane curve; that is, . Moreover, by (52) and (59), we have that . Hence, by (52), . This completes the proof.
From now on, let the binormal vector of the profile curve of be given by such that is scalar for . Now, we will give the important relation between the one of de Sitter principal curvatures and hyperbolic curvature of the profile curve of .
Theorem 23. Let be -invariant surface in . Then, .
Proof. Let the binormal vector of the profile curve of be denoted by . In Section 3, from the definition of Serret-Frenet vectors, we have that . Also, by (50) and (71), we obtain that . Thus, it follows that . For this reason, we have that .
As a result of Theorem 23, the de Sitter Gauss curvature and the de Sitter mean curvature of are at any , respectively. Moreover, if we apply (58), then the hyperbolic Gauss curvature and the hyperbolic mean curvature of are at any , respectively.
Proposition 24. Let , , be unit speed regular profile curve of -invariant surface . Then its components are given by
Proof. Suppose that the profile curve of is unit speed and regular. So that, it satisfies the following equations: for all . By (77) and , we have that such that is a differentiable function. Moreover, by (78) and (79), we obtain that Finally, by (80), we have that such that for all . Without loss of generality, when we choose positive of signature of , this completes the proof.
Remark 25. If is a de Sitter flat surface in , then we say that is an H-plane in .
Now, we will give some results which are obtained by (72) and (74).
Corollary 26. Let be the profile curve of -invariant surface in . Then, (i)if or , then is a part of de Sitter flat surface;(ii)if , then is a part of de Sitter flat surface;(iii)if , then is a part of de Sitter flat surface.
Corollary 27. Let be the profile curve of -invariant surface in . If such that , then is a de Sitter flat surface.
Theorem 28. Let be the profile curve of -invariant surface in . Then, is hyperbolic flat surface if and only if .
Proof. Suppose that is hyperbolic flat surface; that is, . By (74), it follows that or for all . Firstly, let us find solution of (82). If Proposition 24 is applied to (82), we have that . Hence, it follows that There is no real solution of (84). This means that the only one solution is by (83). On the other hand, if we assume that , then the proof is clear.
Corollary 29. Let be the profile curve of -invariant surface in . Then, (i)if and , then is -flat surface which is generated from horocyle;(ii)if and , then is -flat surface which is generated from horocyle.
Now, we will give theorem and corollaries for -invariant surface which satisfy minimal condition in by (73) and (75).
Theorem 30. Let be the profile curve with constant hyperbolic curvature of -invariant surface in . Then, is de Sitter minimal surface if and only if the parametrization of is given by with the condition such that is an arbitrary constant.
Proof. Suppose that is de Sitter minimal surface; that is, . By (73), it follows that
for all . By using Proposition 24, we have the following differential equation:
There exists only one real solution of (87) under the condition . Moreover, must not be zero by Corollary 26. So that, we obtain
Hence, the solution is
where is an arbitrary constant under the condition (88). Finally, the parametrization of is given explicitly by Proposition 24.
On the other hand, let the parametrization of profile curve of be given by (85) under the condition (87). Then it satisfies (86). It means that .
Theorem 31. Let be the profile curve with constant hyperbolic curvature of -invariant surface in . Then, is hyperbolic minimal surface if and only if the parametrization of is given by with the condition such that is an arbitrary constant.
Proof. Suppose that is hyperbolic minimal surface; that is, . By (75), it follows that
If Proposition 24 is applied to (91), we have that
There exists only one real solution of (92) under the condition
Thus, the solution is
where is an arbitrary constant under the condition (93). However, must not be zero by Corollary 26. So that, if is -minimal surface (-minimal surface), then we obtain () by (93). Finally, the parametrization of is given explicitly by Proposition 24.
Conversely, let the parametrization of profile curve of be given by (90) under the condition (93). Then, it satisfies (91). It means that .
Now, we will give classification theorem for totally umbilical -invariant surfaces.
Theorem 32. Let be the profile curve of totally umbilical -invariant surface in . Then, the hyperbolic curvature of is constant.
Proof. Let be totally umbilical -invariant surface. By Theorem 23 and (70), we may assume that for all . Then, we have the following equations: By (95), we obtain that Also, if we use the equations and in (96), then it follows that for all . Moreover, must not be zero by Corollary 26. Thus, is constant.
Corollary 33. Let be the profile curve of totally umbilical -invariant surface in . Then we have the following classification. (1)Supposing that , if and , then is a part of an equidistant surface; if and , then is a part of a sphere; if , then is a part of a H-plane.(2)If , then is a part of horosphere,where is a constant.
Proof. We suppose that . By Proposition 22 and Theorem 32, we have that is constant. Moreover, is de Sitter principal curvature of by Theorem 23. Since is totally umbilical surface, de Sitter shape operator of is where is identity matrix. Finally, the proof is complete by Lemma 20.
Now, we will give some examples of -invariant surface in . Let the Poincaré ball model of hyperbolic space be given by with the hyperbolic metric . Then, it is well known that stereographic projection of is given by We can draw the pictures of surface by using stereographic projection . That is, such that .
Example 34. The -invariant surface which is generated from with hyperbolic curvature is drawn in Figure 1(a).
(a)
(b)
(c)
(d)
Example 35. Let the profile curve of be given by such that hyperbolic curvature . Then, is hyperbolic flat -invariant surface which is generated from horocycle in (see Figure 1(b)).
Example 36. The -invariant surface which is generated from with hyperbolic curvature is drawn in Figure 1(c).
Example 37. Let the profile curve of be given by such that hyperbolic curvature . Then, is totally umbilical -invariant surface with in (see Figure 1(d)).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research of Mahmut Mak was partially supported by the grant of Ahi Evran University Scientific Research Project (PYO.FEN.4009.14.014).