Bour's Theorem in 4-Dimensional Euclidean

Home , Catenoid, Helicoid

Bull. Korean Math. Soc. 54 (2017), No. 6, pp. 2081–2089 https://doi.org/10.4134/BKMS.b160766 pISSN: 1015-8634 / eISSN: 2234-3016

BOUR’S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE

Doan The Hieu and Nguyen Ngoc Thang

Abstract. In this paper we generalize 3-dimensional Bour’s Theorem 4 to the case of 4-dimension. We proved that a helicoidal surface in R 4 is isometric to a family of surfaces of revolution in R in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

1. Introduction Consider the deformation determined by a family of parametric surfaces given by

Xθ(u, v) = (xθ(u, v), yθ(u, v), zθ(u, v)),

xθ(u, v) = cos θ sinh v sin u + sin θ cosh v cos u,

yθ(u, v) = − cos θ sinh v cos u + sin θ cosh v sin u,

zθ(u, v) = u cos θ + v sin θ; where −π < u ≤ π, −∞ < v < ∞ and the deformation parameter −π < θ ≤ π. A direct computation shows that all surfaces Xθ are minimal, have the same ﬁrst fundamental form and normal vector ﬁeld. We can see that X0 is the helicoid and Xπ/2 is the catenoid. Thus, locally the helicoid and catenoid are isometric and have the same Gauss map. Moreover, helices on the helicoid correspond to parallel circles on the catenoid. The classical Bour’s theorem (see [1]) can be seen as a weak generalization of this fact. Bour’s theorem. A helicoidal surface is isometric to a surface of revolution so that helices on the helicoidal surface correspond to parallel circles on the surface of revolution.

Received September 21, 2016; Accepted March 7, 2017. 2010 Mathematics Subject Classiﬁcation. 53A07, 53A10. Key words and phrases. Bour’s theorem, helicoidal surface, surface of revolution, Gauss map, minimal surface.

c 2017 Korean Mathematical Society 2081 2082 D. T. HIEU AND N. N. THANG

Ikawa [7] studied a pair of surfaces by Bour’s theorem in R3 with an additional condition that they have the same Gauss map and proved that they are minimal, i.e they are the helicoid and catenoid, respectively. Bour’s theorem is studied not only in R3, but also in Lorentz-Minkowski 3-space (see [2], [3], [5], [4], [6], [7], [8], [9], [10]). A helicoidal surface in R3 is the orbit of a plane curve under a screw motion, that is a rotation around a line followed by a translation along the line. In R4, we can define the rotation around a plane, therefore we can define surfaces of revolution as well as helicoidal surfaces similarly as those in R3. So it is natural to generalize Bour’s theorem to 4-dimensional case, i.e., the ambient space is R4, and consider the case when the surfaces are required further to have the same Gauss map. The purpose of this paper is to settle these problems. The first main result of the paper is Theorem 3.1, a generalization of the classical Bour’s theorem. The second one is Theorem 3.2 asserting that a pair of surfaces related by Bour’s theorem having the same Gauss map must be hyperplanar, i.e. belong to hyperplanes, and minimal. The parametrizations of such minimal surfaces are determined explicitly in Corollary 3.3.

2. Preliminaries 2.1. Basic concepts

Let X : D → R4,D ⊂ R2 be a smooth parametric surface in R4. The tangent space to M at an arbitrary point p = X(u, v) is the span{Xu,Xv}. The coeﬃcients of the ﬁrst fundamental form of M are given by

g11 = hXu,Xui , g12 = g21 = hXu,Xvi , g22 = hXv,Xvi , 4 2 where h, i is the standard inner product on R . We assume that W = det(gij) = 2 g11g22 − g12 > 0, i.e., the surface M is regular. Let {e1, e2,N1,N2} be a local orthonormal frame field on M such that e1, e2 are tangent to M and N1,N2 are normal to M. The coefficients of the second fundamental form of M with respect to the unit normal vector field Ni, i = 1, 2 are given by i i i i b11 = hXuu,Nii , b12 = b21 = hXuv,Nii , b22 = hXvv,Nii . Denote by i i i b11g22−2b12g12+b22g11 (1) Hi = 2W 2 the mean curvature of M with respect to N , i = 1, 2; −→i (2) H = H1n1 + H2n2 the mean curvature vector of M; 1 1 1 2 2 2 2 2 det(b1 )+det(b )2 b11b22−(b12) +b11b22−(b12) ij ij (3) K = W 2 = W 2 the Gauss curvature of M. A surface M is said to be minimal if its mean curvature vector vanishes identically. Denote G(2, 4) the Grassmann manifold consisting of all oriented 2-planes passing through the origin in R4. The map G : M → G(2, 4) that assigns each BOUR’S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE 2083 point of M to its oriented tangent space in R4 is called the Gauss map of the surface M. We can identify naturally an oriented 2-planes passing through the 4 4 V2 4 origin in R with a unit simple 2-vector in R , i.e., an element of R . The V2 4 space R is a 6-dimensional Euclidean space with the canonical orthonormal basic {i ∧ j : i < j; i, j = 1, 2, 3, 4}, where {1, 2, 3, 4, } is the canonical basis of R4. For more details about the space of k-vectors, we refer the reader to Chapter 4 of [11]. We can choose an orthonormal tangent frame field {e1, e2} on M as follows 1 1 e1 = √ Xu, e2 = √ (g11Xv − g12Xu). g11 W g11 and Gauss map of M can be written as 1 G = X ∧ X . W u v In the rest of this paper, sometime a vector (a, b, c, d) ∈ R4 is identified with its transpose, i.e., a column matrix.

2.2. Helicoidal surfaces in R4 Let I be an interval in R, Π be a hyperplane in R4,P ⊂ Π be a 2-plane and c : I → Π be a smooth parametric curve in Π, such that c(I) ∩ P = ∅. The orbit of the curve c under the rotation around the plane P is a surface of revolution in R4. When c rotates around the plane P , it simultaneously translates along a line l parallel to P in such a way that the speed of the translation is proportional to the speed of the rotation, then the resulting surface is a helicoidal surface in R4. Let x, y, z, w be the coordinates in R4. By a suitable change of coordinates, we may assume that Π is the xzw-hyperplane, P is zw-plane, l is parallel to the z-axis. Now suppose that c(u) = (u, 0, f(u), g(u)), then a parametrization of the helicoidal surface H generated by c is (1) H(u, v) = (u cos v, u sin v, f(u) + av, g(u)), where u > 0, 0 ≤ v < 2π and a > 0. When g is a constant function, then the surface is just a helicoidal surface in E3. A direct computation yields 0 0 Hu = (cos v, sin v, f , g ) , Hv = (−u sin v, u cos v, a, 0) , and 02 02 0 2 2 g11 = 1 + f + g , g12 = g21 = af , g22 = u + a .

We consider the following orthonormal frame ﬁeld {e1, e2,N1,N2} on H such that e1, e2 are tangent to H and N1,N2 are normal to H 1 1 e1 = √ Hu, e2 = √ (g11Hv − g12Hu), g11 W g11 2084 D. T. HIEU AND N. N. THANG

1 0 0 N1 = (g cos v, g sin v, 0, −1), p1 + g02  a(1 + g02) sin v − uf 0 cos v  1  −a(1 + g02) cos v − uf 0 sin v  N2 =  02  , W p1 + g02  u(1 + g )  −uf 0g0

p 2 p 2 2 02 2 02 where W = det(gij) = g11g22 − g12 = (u + a )(1 + g ) + u f 6= 0. The coeﬃcients of the second fundamental form, the Gauss curvature K and the mean curvatures Hi (i = 1, 2) are 00 0 1 −g 1 1 −ug b11 = , b12 = 0, b22 = , p1 + g02 p1 + g02 00 02 0 0 00 02 2 0 2 uf (1 + g ) − uf g g 2 −a(1 + g ) 2 u f b11 = , b12 = , b22 = , W p1 + g02 W p1 + g02 W p1 + g02

(u2 + a2)ug0g00 + u3f 0f 00 − a2(1 + g02) (2) K = , W 4 −(u2 + a2)g00 − ug0(1 + f 02 + g02) H1 = , 2W 2p1 + g02 (3) u(u2 + a2) f 00(1 + g02) − f 0g0g00 + 2a2f 0(1 + g02) + u2f 0(1 + f 02 + g02) H2 = ; 2W 3p1 + g02 respectively.

3. Generalized Bour’s theorem The following theorem is a generalization of the classical Bour’s theorem.

Theorem 3.1. In R4, let H be a helicoidal surface given by (1). Suppose that k(u), l(u), u > 0, are diﬀerentiable functions satisfying the following ODE a2 + u2f 02 + (u2 + a2)g02 (4) k2 + l2 = . u2 Then the helicoidal surface H is isometric to a family of surfaces of revolution determined by √ 0  2 2 R af  u + a cos v + u2+a2 du  √ 0   u2 + a2 sin v + R af du  (5) R(u, v) =  u2+a2  ,  u k(u)  R √ du  u2+a2   u l(u)  R √ du u2+a2 in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. BOUR’S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE 2085

Proof. The first fundamental form of the helicoidal surface (1) is given by ds2 = (1 + f 02 + g02) du2 + 2af 0 dudv + (u2 + a2) dv2. R af 0 Set u = u, v = v + u2+a2 du. Since the Jacobian ∂(u, v)/∂(u, v) is non-zero, it follows that (u, v) are new parameters for H. In terms of these new parameters, the first fundamental form becomes ! u2f 02 ds2 = 1 + + g02 du¯2 + u2 + a2 dv¯2. u2 + a2 On the other hand, the first fundamental form of a surface of revolution in R4 (6) R(w, t) = (w cos t, w sin t, ϕ(w), ψ(w)) is ds2 = (1 + ϕ02 + ψ02) dw2 + w2 dt2. √ Set w = u2 + a2 and k(u) = ϕ0, l(u) = ψ0, we obtain Z u k(u) Z u l(u) ϕ = √ du, ψ = √ du, u2 + a2 u2 + a2 and therefore the required ODE (4) implies that the helicoidal surface (1) is isometric to the surface of revolution (5). It is easy to see that, a helix on H is defined by u = u0, where u0 is a p 2 2 constant, and it corresponds to the curves on R defined by w = u0 + a , i.e., circles on the plane {x3 = ϕ(w), x4 = ψ(w)}. Next we consider the isometric surfaces in Theorem 3.1 with an additional condition that they have the same Gauss map. Theorem 3.2. Let H and R be a helicoidal surface and a surface of revolution that are isometrically related by Theorem 3.1. If the surfaces have the same Gauss map, then they are hyperplanar and minimal.

4 Proof. Let {e1, e2, e3, e4} be the standard orthonormal basis in R and denote eij := ei ∧ ej, i, j = 1, 2, 3, 4, i < j. Then, the Gauss map of the helicoidal surface (1) is 1 G = (H ∧ H ) H W u v 1 = u e + (a cos v + uf 0 sin v) e + ug0 sin v e W 12 13 14 0 0 0 (7) + (a sin v − uf cos v) e23 − ug cos v e24 − ag e34 , while the Gauss map of the surface of revolution (5) is

1 0 0 G = u e + uk sin v + R af du e + ul sin v + R af du e R W 12 u2+a2 13 u2+a2 14 2086 D. T. HIEU AND N. N. THANG R af 0 R af 0 (8) − uk cos v + u2+a2 du e23 − ul cos v + u2+a2 du e24 .

If GH is identically equal to GR, then (7) and (8) yield Z af 0 (9) a cos v + uf 0 sin v = uk sin v + du , u2 + a2 Z af 0 (10) a sin v − uf 0 cos v = −uk cos v + , u2 + a2 Z af 0 (11) ug0 sin v = ul sin v + du , u2 + a2 Z af 0 (12) −ug0 cos v = −ul cos v + du , u2 + a2 (13) −ag0 = 0. From (11), (12) and (13) we obtain g0 = 0, l = 0; i.e., H and R are hyperplanar. We will prove that they are minimal (see [7] for the proof of 3-dimensional case). Because l = 0, k 6= 0 by (4). Multiplying (9) by cos v, (10) by sin v and then summing them up, we obtain Z af 0 (14) a = uk sin du . u2 + a2 Multiplying (9) by sin v, (10) by cos v and then subtracting the ﬁrst from the later one, we obtain Z af 0 (15) uf 0 = uk cos du . u2 + a2 Thus, uf 0 Z af 0 = cot du , a u2 + a2 or uf 0 Z af 0 (16) cot−1 = du. a u2 + a2 Taking the derivative both sides of (16), we obtain (17) (u2 + a2)uf 00 + u2f 0(1 + f 02) + 2a2f 0 = 0. The mean curvatures of the helicoidal surface with respect to 0  a sin v − uf 0 cos v  0 H 0 H −a cos v − uf sin v N =   and N =   1 0 2  u  1 0 BOUR’S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE 2087 are 2 2 00 2 0 02 2 0 H H (u + a )uf + u f (1 + f ) + 2a f H1 = 0, and H2 = . 2 (a2 + u2 + u2f 02)3/2 The mean curvatures of the surface of revolution with respect to 0 R 0 N =   and 1 0 1  0  p 2 2 02 R af − a + u f cos v + u2+a2 du  p af 0  R 1 − a2 + u2f 02 sin v + R du  N2 =  u2+a2  pa2 + u2 + u2f 02    u  0 are 2 0 2 2 00 2 0 02 2 0 R R u f (u + a )uf + u f (1 + f ) + 2a f H1 = 0 and H2 = √ . 2 u2 + a2pa2 + u2f 02 (a2 + u2 + u2f 02)3/2 Therefore, both H and R are minimal by (17). Corollary 3.3. Let H and R be a helicoidal surface and a surface of revolution having the same Gauss map that are isometrically related by Theorem 3.1. Then their parametrizations can be determined explicitly as follows.

H(u, v) = (u cos v, u sin v, f(u) + av, c1),

 √ 0  2 2 R af u + a cos v + u2+a2 du  √ 0   u2 + a2 sin v + R af du   u2+a2  R(u, v) =  √  ,  b cosh−1 u2+a2   b  c2 where

√ √ √ p q 2 2 2 2 2 2 2 q 2 2 f(u) = b2 − a2 ln √u +a +√u +a −b − a arctan b −a u +a , u2+a2− u2+a2−b2 a u2+a2−b2 √ 2 2 and c1, c2, b are constants, b ≥ a and u > b − a . Proof. By the above proof, H and R are hyperplanar. Suppose that H belongs to the hyperplane w = c1 and R belongs to the hyperplane w = c2. Because R −1 w is minimal, it must be the catenoid and therefore ϕ(w) = b cosh b , where b is a nonzero constant. We can assume b is positive. Thus, √ ! s u2 + a2 Z a2 + u2f 02 b cosh−1 = du. b u2 + a2 2088 D. T. HIEU AND N. N. THANG

It follows that √ √ b2 − a2 u2 + a2 (18) f 0 = √ . u u2 + a2 − b2

q u2+a2 Set t = u2+a2−b2 > 0, we have p Z −b2t2 f = b2 − a2 dt (t2 − 1)((b2 − a2)t2 + a2) r √ p t + 1 b2 − a2 = b2 − a2 ln − a arctan t. t − 1 a Remark 3.4. If f 0 = 0, then the helicoidal surface reduces to the helicoid and the mean curvature vector of rotation surface is identically zero, i.e., it is the catenoid. In this case, by (18), it follows that a = b. Acknowledgements. This research is funded by Vietnam National Foun- dation for Science and Technology Development (NAFOSTED) under grant number 101.04.2014.26.

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Doan The Hieu College of Education Hue University 32 Le Loi, Hue, Vietnam E-mail address: [email protected] BOUR’S THEOREM IN 4-DIMENSIONAL EUCLIDEAN SPACE 2089

Nguyen Ngoc Thang College of Education Hue University 32 Le Loi, Hue, Vietnam