The Second Laplace-Beltrami Operator on Rotational Hypersurfaces in the Euclidean 4-Space

Transcription

1 Mathematica Aeterna, Vol. 8, 218, no. 1, 1-12 The Second Laplace-Beltrami Operator on Rotational Hypersurfaces in the Euclidean 4-Space Erhan GÜLER and Ömer KİŞİ Bartın University, Faculty of Sciences Department of Mathematics, 741 Bartın, Turkey Abstract We consider rotational hypersurface in the four dimensional Euclidean space. We calculate the mean curvature and the Gaussian curvature, and some relations of the rotational hypersurface. Moreover, we define the second Laplace-Beltrami operator and apply it to the rotational hypersurface. Mathematics Subject Classification: 53A35 Keywords: The Second Laplace-Beltrami operator, Rotational hypersurface, Gaussian curvature, mean curvature, 4-space 1 Introduction The notion of finite type immersion of submanifolds of a Euclidean space has been used in classifying and characterizing well known Riemannian submanifolds [4]. Chen [4] posed the problem of classifying the finite type surfaces in the 3-dimensional Euclidean space E 3. A Euclidean submanifold is said to be of Chen finite type if its coordinate functions are a finite sum of eigenfunctions of its Laplacian. Further, the notion of finite type can be extended to any smooth function on a submanifold of a Euclidean space or a pseudo-euclidean space. Then the theory of submanifolds of finite type has been studied by many geometers. Takahashi [22] states that minimal surfaces and spheres are the only surfaces in E 3 satisfying the condition r = λr, λ R. Ferrandez, Garay and Lucas [1] prove that the surfaces of E 3 satisfying H = AH, A Mat3, 3 are either minimal, or an open piece of sphere or of a right circular cylinder.

2 2 Erhan Güler and Ömer Kişi Choi and Kim [6] characterize the minimal helicoid in terms of pointwise 1-type Gauss map of the first kind. Dillen, Pas and Verstraelen [8] prove that the only surfaces in E 3 satisfying r = Ar + B, A Mat3, 3, B Mat3, 1 are the minimal surfaces, the spheres and the circular cylinders. Senoussi and Bekkar [21] study helicoidal surfaces M 2 in E 3 which are of finite type in the sense of Chen with respect to the fundamental forms I, II and III, i.e., their position vector field ru, v satisfies the condition J r = Ar, J = I, II, III, A = a ij is a constant 3 3 matrix and J denotes the Laplace operator with respect to the fundamental forms I, II and III. When we focus on the ruled helicoid and rotational characters, we see Bour s theorem in [3]. About helicoidal surfaces in Euclidean 3-space, do Carmo and Dajczer [9] prove that, by using a result of Bour [3], there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface. Güler [12] studies on a helicoidal surface with lightlike profile curve using Bour s theorem in Minkowski geometry. Also, Hieu and Thang [13] study helicoidal surfaces by Bour s theorem in 4-space. Choi et al. [7] study on helicoidal surfaces and their Gauss map in Minkowski 3-space. Kim and Turgay [14] classify the helicoidal surfaces with L 1 -pointwise 1-type Gauss map. Lawson [15] gives the general definition of the Laplace-Beltrami operator in his lecture notes. Magid, Scharlach and Vrancken [16] introduce the affine umbilical surfaces in 4-space. Vlachos [24] consider hypersurfaces in E 4 with harmonic mean curvature vector field. Scharlach [2] studies the affine geometry of surfaces and hypersurfaces in 4-space. Cheng and Wan [5] consider complete hypersurfaces of 4-space with constant mean curvature. Arvanitoyeorgos, Kaimakamais and Magid [2] show that if the mean curvature vector field of M1 3 satisfies the equation H = αh α a constant, then M1 3 has constant mean curvature in Minkowski 4-space E1. 4 This equation is a natural generalization of the biharmonic submanifold equation H =. General rotational surfaces as a source of examples of surfaces in the four dimensional Euclidean space were introduced by Moore [17, 18]. Arslan et al [1] study on generalized rotation surfaces in E 4. Ganchev and Milousheva [11] consider the analogue of these surfaces in the Minkowski 4-space. They classify completely the minimal general rotational surfaces and the general rotational surfaces consisting of parabolic points. Moruz and Munteanu [19] consider hypersurfaces in the Euclidean space E 4 defined as the sum of a curve and a surface whose mean curvature vanishes. They call them minimal translation hypersurfaces in E 4 and give a classification of these hypersurfaces. Verstraelen, Walrave and Yaprak [23] study the minimal translation surfaces in E n for arbitrary dimension n. We consider the rotational hypersurfaces in Euclidean 4-space E 4 in this

3 The Second Laplace-Beltrami Operator... 3 paper. We give some basic notions of the four dimensional Euclidean geometry in Section 2. In Section 3, we give the definition of a rotational hypersurface. We calculate the mean curvature and the Gaussian curvature of the rotational hypersurface. We introduce the second Laplace-Beltrami operator in Section 4. Moreover, we calculate the second Laplace-Beltrami operator of the rotational hypersurface in E 4 in the last section. 2 Preliminaries In this section, we will introduce the first and second fundamental forms, matrix of the shape operator S, Gaussian curvature K, and the mean curvature H of hypersurface M = Mr, θ 1, θ 2 in Euclidean 4-space E 4. In the rest of this work, we shall identify a vector α with its transpose. Let M = Mr, θ 1, θ 2 be an isometric immersion of a hypersurface M 3 in the E 4. The inner product of x = x 1, x 2, x 3, x 4, y = y 1, y 2, y 3, y 4 on E 4 is defined by as follows: x y = x1 y 1 + x 2 y 2 + x 3 y 3 + x 4 y 4. The vector product of x = x 1, x 2, x 3, x 4, y = y 1, y 2, y 3, y 4, z = z 1, z 2, z 3, z 4 on E 4 is defined by as follows: e 1 e 2 e 3 e 4 x y z = det x 1 x 2 x 3 x 4 y 1 y 2 y 3 y 4. z 1 z 2 z 3 z 4 For a hypersurface Mr, θ 1, θ 2 in 4-space, we have E F A det I = det F G B = EG F 2 C A 2 G + 2ABF B 2 E, 1 A B C and det II = det L M P M N T P T V = LN M 2 V P 2 N + 2P T M T 2 L, 2 I and II are the first and the second fundamental form matrices, respectively, E = M r M r, F = M r M θ1, G = M θ1 M θ1, A = M r M θ2, B = M θ1 M θ2, C = M θ2 M θ2, L = M rr e, M = M rθ1 e, N = M θ1 θ 1 e,

4 4 Erhan Güler and Ömer Kişi P = M rθ2 e, T = M θ1 θ 2 e, V = M θ2 θ 2 e. Here, e is the Gauss map i.e. the unit normal vector defined by Product matrices e = E F A F G B A B C M r M θ1 M θ2 M r M θ1 M θ L M P M N T P T V gives the matrix of the shape operator S as follows: S = 1 s 11 s 12 s 13 s 21 s 22 s 23, 4 det I s 31 s 32 s 33 s 11 = ABM CF M AGP + BF P + CGL B 2 L, s 12 = ABN CF N AGT + BF T + CGM B 2 M, s 13 = ABT CF T AGV + BF V + CGP B 2 P, s 21 = ABL CF L + AF P BP E + CME A 2 M, s 22 = ABM CF M + AF T BT E + CNE A 2 N, s 23 = ABP CF P + AF V BV E + CT E A 2 T, s 31 = AGL + BF L + AF M BME + GP E F 2 P, s 32 = AGM + BF M + AF N BNE + GT E F 2 T, s 33 = AGP + BF P + AF T BT E + GV E F 2 V. The formulas of the Gaussian and the mean curvatures, respectively as follow: det II K = dets = det I, 5 and H = 1 tr S, 3 6 tr S = 1 det I [EN + GL 2F MC + EG F 2 V A 2 N B 2 L 2AP G + BT E ABM AT F BP F ]. A hypersurface M is minimal if H = identically on M.

5 The Second Laplace-Beltrami Operator Curvatures of a rotational hypersurface We define the rotational hypersurface in E 4. For an open interval I R, let γ : I Π be a curve in a plane Π in E 4, and let l be a straight line in Π. A rotational hypersurface in E 4 is hypersurface rotating a curve γ around a line l these are called the profile curve and the axis, respectively. We may suppose that l is the line spanned by the vector,,, 1 t. The orthogonal matrix which fixes the above vector is Zθ 1, θ 2 = cos θ 1 cos θ 2 sin θ 1 cos θ 1 sin θ 2 sin θ 1 cos θ 2 cos θ 1 sin θ 1 sin θ 2 sin θ 2 cos θ 2 1, 7 θ 1, θ 2 R. The matrix Z can be found by solving the following equations simultaneously; Zl = l, Z t Z = ZZ t = I 4, det Z = 1. When the axis of rotation is l, there is an Euclidean transformation by which the axis is l transformed to the x 4 -axis of E 4. Parametrization of the profile curve is given by γr = r,,, ϕ r, ϕ r : I R R is a differentiable function for all r I. So, the rotational hypersurface which is spanned by the vector,,, 1, is as follows: Rr, θ 1, θ 2 = Zθ 1, θ 2.γr t 8 in E 4, r I, θ 1, θ 2 [, 2π]. When θ 2 =, we have rotational surface in E 4. r cos θ 1 cos θ 2 Rr, θ 1, θ 2 = r sin θ 1 cos θ 2 r sin θ 2. 9 ϕr r R\{} and θ 1, θ 2 2π. Next, we obtain the mean curvature and the Gaussian curvature of the rotational hypersurface 9. The first differentials of 9 with respect to r, θ 1, θ 2, respectively, we get the first quantities of 9 as follow: I = 1 + ϕ 2 r 2 cos 2 θ 2 r 2. 1

6 6 Erhan Güler and Ömer Kişi We also have det I = r ϕ 2 cos 2 θ 2, ϕ = ϕr, ϕ = dϕ. Using 3, we get the Gauss map of the rotational dr hypersurface 9 as follows r 2 ϕ cos θ 1 cos 2 θ 2 e R = 1 r 2 ϕ sin θ 1 cos 2 θ 2 det I r 2 ϕ sin θ 2 cos θ r 2 cos θ 2 The second differentials of 9 with respect to r, θ 1, θ 2, respectively, we get r cos θ 1 cos θ 2 r cos θ 1 cos θ 2 R rr =, R θ 1 θ 1 = r sin θ 1 cos θ 2, R θ 2 θ 2 = r sin θ 1 cos θ 2 r sin θ 2. ϕ Using the second differentials above and the Gauss map 11 of the rotational hypersurface 9, we have the second quantities as follow: II = 1 r 2 ϕ cos θ 2 r 3 ϕ cos 3 θ det I r 3 ϕ cos θ 2 So, we have det II = r2 ϕ 2 ϕ cos 2 θ ϕ 2 3/2. We calculate the shape operator matrix of the rotational hypersurface 9, using 4, as follows: ϕ 1+ϕ 2 3/2 S = ϕ r1+ϕ 2 1/2. ϕ r1+ϕ 2 1/2 Finally, using 5 and 6, respectively, we calculate the Gaussian curvature and the mean curvature of the rotational hypersurface 9 as follow: ϕ 2 ϕ K = and H = rϕ + 2ϕ 3 + 2ϕ r ϕ 2 5/2 3r 1 + ϕ 2. 3/2 Corollary 1. Let R : M 3 E 4 be an isometric immersion given by 9. Then M 3 has constant Gaussian curvature if and only if ϕ 4 ϕ 2 Cr ϕ 2 5 =.

7 The Second Laplace-Beltrami Operator... 7 Corollary 2. Let R : M 3 E 4 be an isometric immersion given by 9. Then M 3 has constant mean curvature CMC if and only if rϕ + 2ϕ 3 + 2ϕ 2 9Cr ϕ 2 3 =. Corollary 3. Let R : M 3 E 4 be an isometric immersion given by 9. Then M 3 has zero Gaussian curvature if and only if ϕr = c 1 r + c 2. Proof. Solving the 2nd order differential eq. K =, i.e. ϕ 2 ϕ =, we get the solution. Corollary 4. Let R : M 3 E 4 be an isometric immersion given by 9. Then M 3 has zero mean curvature if and only if ϕr = ± c 1 dr r4 c 1 + c 2 = EllipticF Ir, I, EllipticF Ir, I is incomplete elliptic integral of the first kind. Proof. When we solve the 2nd order differential eq. H =, i.e. rϕ + 2ϕ 3 + 2ϕ =, we get the solution. 4 The second Laplace-Beltrami operator The inverse of the matrix h ij = h 11 h 12 h 13 h 21 h 22 h 23 h 31 h 32 h 33 is as follows: h 1 22 h 33 h 23 h 32 h 12 h 33 h 13 h 32 h 12 h 23 h 13 h 22 h 21 h 33 h 31 h 23 h 11 h 33 h 13 h 31 h 11 g 23 h 21 h 13 h h 21 h 32 h 22 h 31 h 11 h 32 h 12 h 31 h 11 h 22 h 12 h 21 h = det h ij = h 11 h 22 h 33 h 11 h 23 h 32 + h 12 h 31 h 23 h 12 h 21 h 33 + h 21 h 13 h 32 h 13 h 22 h 31.,

8 8 Erhan Güler and Ömer Kişi The second Laplace-Beltrami operator of a smooth function φ = φx 1, x 2, x 3 D D R 3 of class C 3 with respect to the second fundamental form of hypersurface M is the operator which is defined by as follows: II φ = 1 X 3 hh ij φ h i,j=1 x i x j h ij = h kl 1 and h = det h ij. Clearly, we write II φ as follows: 1 h hh 11 φ hh 12 φ hh + 13 φ x 1 x 1 x 1 x 2 x 1 x hh 21 φ hh + 22 φ hh 23 φ x 2 x 1 x 2 x 2 x 2 x 3 hh + 31 φ hh 32 φ hh + 33 φ x 3 x 1 x 3 x 2 x 3 x 3 So, using more clear notation, we get,. 14 II 1 = 1 det II NV T 2 P T MV MT NP P T MV LV P 2 MP LT MT NP MP LT LN M 2, det II = LN M 2 V P 2 N + 2P T M T 2 L. Hence, using more transparent notation we get the second Laplace-Beltrami operator of a smooth function φ = φr, θ 1, θ 2 as follow: II φ = 1 NV T 2 φ r P T MV φ θ1 +MT NP φ θ2 r P T MV φ r LV P 2 φ θ1 +MP LT φ θ2 θ 1 + MT NP φ r MP LT φ θ1 +LN M 2 φ θ2 θ We continue our calculations to find the second Laplace-Beltrami operator II R of the rotational hypersurface R using 15 to the 9. The second Laplace-Beltrami operator of the hypersurface 9 is given by II R = 1 r U V + W, θ 1 θ 2

9 The Second Laplace-Beltrami Operator... 9 U = NV T 2 R r P T MV R θ1 + MT NP R θ2, V = P T MV R r LV P 2 R θ1 + MP LT R θ2, W = MT NP R r MP LT R θ1 + LN M 2 R θ2. Hence, using the hypersurface 9, we get M = P = T =. Therefore, we write U, V, W again, as follow: NV r 2 ϕ 2 cos 2 θ U = 2 R r = 1 + ϕ 2 R r, V = LV rϕ ϕ R θ1 = 1 + ϕ 2 R θ1, LN rϕ ϕ cos 2 θ W = 2 R θ2 = 1 + ϕ 2 R θ2, det II = r2 ϕ 2 ϕ cos 2 θ ϕ 2 3/2. Using differentials of r, θ 1, θ 2 on U, V, W, respectively, we get r U = cos θ 1 cos 2 θ 2 rϕ sin θ 1 cos 2 θ 2 r ϕ 1 + ϕ 2 1/4 sin θ 2 cos θ 2 ϕ Hence, we have θ 1 V = θ 2 W = r ϕ 1 + ϕ 2 1/4 ϕ 1 + ϕ 2 1/4 + rϕ ϕ 1 + ϕ 2 1/4 cos θ 1 sin θ 1, r cos θ 1 cos 2θ 2 r sin θ 1 cos 2θ 2 r sin 2θ 2 II R = II R 1, II R 2, II R 3, II R 4,. ϕ,

10 1 Erhan Güler and Ömer Kişi II R 1 = δ.[ 1 2 ϕ 1 2 ϕ 2 ϕ +rϕ 2 ϕ 2 +rϕ 2 +ϕ ϕ ϕ +ϕ 3 ϕ cos θ 1 cos 2 θ 2 rϕ cos θ cos θ 1 cos 2θ 2 rϕ 2 ϕ cos θ cos θ 1 cos 2θ 2 ], II R 2 = δ.[ 1 2 ϕ 1 2 ϕ 2 ϕ +rϕ 2 ϕ 2 +rϕ 2 +ϕ ϕ ϕ +ϕ 3 ϕ sin θ 1 cos 2 θ 2 rϕ sin θ sin θ 1 cos 2θ 2 rϕ 2 ϕ sin θ sin θ 1 cos 2θ 2 ], II R 3 = δ.[ 1 2 ϕ 1 2 ϕ 2 ϕ +rϕ 2 ϕ 2 +rϕ 2 +ϕ ϕ ϕ +ϕ 3 ϕ sin θ 2 cos θ 2 rϕ sin 2θ 2 rϕ 2 ϕ sin 2θ 2 ], II R 4 = δ.[ ϕ 2 ϕ ϕ + rϕ 3 ϕ + rϕ ϕ ϕ + ϕ 2 + ϕ 4 + rϕ + rϕ 3 ϕ ], δ = rϕ ϕ ϕ 2 cos θ 2. Remark 1. When the rotational hypersurface R has the equation II R =, i.e. the rotational hypersurface 9 is II minimal, then we have to solve the system of eq. as follow: II R i =, 1 i 4. Here, finding the function ϕ is a future problem for us. Corollary 5. Here ϕ c = const. or ϕ c 1 r + c 2, and θ 2 π 2 + 2kπ, k Z, then we have II R. Hence, the rotational hypersurface 9 is not II minimal. ACKNOWLEDGEMENTS. This work has supported by Bartın University Scientific Research Projects Commision Project Number: BAP FEN - A - 7. References [1] Arslan K., Kılıç Bayram B., Bulca B., Öztürk G. Generalized Rotation Surfaces in E 4. Result Math [2] Arvanitoyeorgos A., Kaimakamis G., Magid M. Lorentz hypersurfaces in E 4 1 satisfying H = αh. Illinois J. Math , [3] Bour E. Théorie de la déformation des surfaces. J. de Polytechnique l.êcole Imperiale [4] Chen B.Y. Total mean curvature and submanifolds of finite type. World Scientific, Singapore, [5] Cheng Q.M., Wan Q.R. Complete hypersurfaces of R 4 with constant mean curvature. Monatsh. Math , [6] Choi M., Kim Y.H. Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc

11 The Second Laplace-Beltrami Operator [7] Choi M., Kim Y.H., Liu H., Yoon D.W. Helicoidal surfaces and their Gauss map in Minkowski 3-space. Bull. Korean Math. Soc , [8] Dillen F., Pas J., Verstraelen L. On surfaces of finite type in Euclidean 3-space. Kodai Math. J [9] Do Carmo M., Dajczer M. Helicoidal surfaces with constant mean curvature. Tohoku Math. J [1] Ferrandez A., Garay O.J., Lucas P. On a certain class of conformally at Euclidean hypersurfaces. Proc. of the Conf. in Global Analysis and Global Differential Geometry, Berlin, 199. [11] Ganchev G., Milousheva, V. General rotational surfaces in the 4- dimensional Minkowski space. Turkish J. Math [12] Güler, E. Bour s theorem and lightlike profile curve, Yokohama Math. J , [13] Hieu D.T., Thang, N.N. Bour s theorem in 4-dimensional Euclidean space. Bull. Korean Math. Soc [14] Kim Y.H., Turgay N.C. Classifications of helicoidal surfaces with L 1 - pointwise 1-type Gauss map. Bull. Korean Math. Soc [15] Lawson H.B. Lectures on minimal submanifolds. Vol. 1, Rio de Janeiro, [16] Magid M., Scharlach C., Vrancken L. Affine umbilical surfaces in R 4. Manuscripta Math [17] Moore C. Surfaces of rotation in a space of four dimensions. Ann. Math [18] Moore C. Rotation surfaces of constant curvature in space of four dimensions. Bull. Amer. Math. Soc [19] Moruz M., Munteanu M.I. Minimal translation hypersurfaces in E 4. J. Math. Anal. Appl [2] Scharlach, C. Affine geometry of surfaces and hypersurfaces in R 4. Symposium on the Differential Geometry of Submanifolds, France

12 12 Erhan Güler and Ömer Kişi [21] Senoussi B., Bekkar M. Helicoidal surfaces with J r = Ar in 3- dimensional Euclidean space. Stud. Univ. Babeş-Bolyai Math [22] Takahashi T. Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan [23] Verstraelen L., Walrave J., Yaprak S. The minimal translation surfaces in Euclidean space. Soochow J. Math [24] Vlachos Th. Hypersurfaces in E 4 with harmonic mean curvature vector field. Math. Nachr Received: January 5, 218