Non-Degenerate Quadric Surfaces in Euclidean 3-Space

Transcription

1 Int. Journal of Math. Analysis, Vol. 6, 2012, no. 52, Non-Degenerate Quadric Surfaces in Euclidean 3-Space Dae Won Yoon and Ji Soon Jun Department of Mathematics Education and RINS Gyeongsang National University, Jinju , South Korea Abstract. In this paper, we study quadric surfaces in a Euclidean 3-space. Furthermore, we classify non-degenerate quadric surfaces in a Euclidean 3- space in terms of the isometric immersion and the Gauss map. Mathematics Subject Classification: 53A05, 53B25 Keywords: Laplacian, quadric surface, ruled surface, second fundametal form 1. Introduction Let x : M E 3 be an isometric immersion of a surface in a Euclidean 3-space. Denote by H and Δ respectively the mean curvature vector field and the Laplacian operator with respect to the induced metric on M induced from that of E 3. Then, as is well known (1.1) Δx = 2H. In [9] T. Takahashi proved that minimal surfaces and spheres are the only surfaces in E 3 satisfying the condition (1.2) Δx = λx, λ R, and O. J. Garay ([4]) extended it to the hypersurfaces, that is, he studied the hypersurfaces in E n+1 on which (1.3) Δx = Ax, A Mat(n +1, R) is satisfied, where Mat(n +1, R) is the set of (n +1) (n + 1)-real matrices. On the other hand, the theory of Gauss map is always one of interesting topics in a Euclidean space and it has been investigated from the various viewpoints by many differential geometers ([2], [3], [5], [7], [8]). It is well known that M has constant mean curvature if and only if ΔG = dg 2 G, where G is

2 2556 Dae Won Yoon and Ji Soon Jun the Gauss map of M ([8]). As a special case one can consider surfaces whose Gauss map G is an eigenfunction of the Laplacian operator, that is, (1.4) ΔG = λg, λ R. B.-Y. Chen and P. Piccinni ([2]) proved that the only compact surface in a Euclidean 3-space satisfying (1.4) is a sphere. C. Jang ([5]) studied that an orientable, connected surface in a Euclidean 3-space satisfying (1.4) is a sphere or a circular cylinder. On the generalization of (1.4), F. Dillen, J. Pas and L. Verstraelen ([3]) studied surfaces of revolution in a Euclidean 3-space E 3 such that their Gauss map G satisfies the condition (1.5) ΔG = AG, A Mat(3, R) and proved that such surfaces are part of the planes, the spheres and the circular cylinders. If a surface M in E 3 has non-degenerate second fundamental form II, then the second fundamental form II is regarded as a new (pseudo-)riemannian metric. So, considering the conditions (1.2) and (1.4), we have the following question: Classify all surfaces with non-degenerate second fundamental form II in a Euclidean 3-space satisfying the conditions (1.6) Δ II x = λx, (1.7) Δ II G = λg, where Δ II is the Laplacian operator with respect to II of the surface. In this paper, we would like to contribute the solution of the above question, by studying this question for a quadric surface in a Euclidean 3-space E 3. Recently, in [10] authors investigated the properties of (non-degenerate) quadric surfaces in a Euclidean 3-space in terms of the curvatures. 2. Preliminaries We denote a surface M in a Euclidean 3-space E 3 by x(u, v) =(x 1 (u, v),x 2 (u, v),x 3 (u, v)). Let n be the standard unit normal vector field on a surface M defined by n = xu xv, where x x u x v u = x(u,v). Then the first fundamental form I of a u surface M is defined by I = g 11 du 2 +2g 12 dudv + g 22 dv 2, where g 11 = x u, x u, g 12 = x u, x v, g 22 = x v, x v. We define the second fundamental form II of M by II = h 11 du 2 +2h 12 dudv + h 22 dv 2 where h 11 = x uu, n, h 12 = x uv, n, h 22 = x vv, n.

3 Non-degenerate quadric surfaces 2557 Now, we define a quadric surface in E 3. A subset M of a Euclidean 3-space E 3 is called a quadric surface if it is the set of points (x 1,x 2,x 3 ) satisfying the following equation of the second degree: 3 3 (2.1) a ij x i x j + b i x i + c =0, i,j=1 i=1 where a ij,b i,c are all real numbers. Suppose that M is not a plane. Then A is not a zero matrix and we may assume without loss of generality that the matrix A =(a ij ) is symmetric. By applying a coordinate transformation in E 3 if necessary, M is either a ruled surface, or one of the following two kinds ([1]) (2.2) or x 2 3 ax 2 1 bx 2 2 = c, abc 0 (2.3) x 3 = a 2 x2 1 + b 2 x2 2, a > 0,b>0. If a surface satisfies the equation (2.2), it is said to be a quadric surface of the first kind M 1 and we call a surface satisfying (2.3) a quadric surface of the second kind M 2. Let {u 1,u 2 } be a local coordinate system of M. For the components h ij (i, j = 1, 2) of non-degenerate second fundamental form II on M we denote by (h ij ) (resp. H) the inverse matrix (resp. the determinant) of the matrix (h ij ). The Laplacian operator Δ II of the second fundamental form II on M is formally defined by (cf. [6]) (2.4) Δ II = 1 H 2 i,j=1 ( H h ij ). u i u j 3. Laplacian of the second fundamental form of M 1 and M 2 In this section, we investigate quadric surfaces of the first kind and the second kind with non-degenerate second fundamental form in E 3. Firstly, we consider a quadric surface of the first kind M 1 in E 3. In this case, M 1 is parametrized by (3.1) x(u, v) =(u, v, (c + au 2 + bv 2 ) 1 2 ). Let s denote the function c + au 2 + bv 2 by ω. Then, the first and the second fundamental form are given by, respectively, I =(1+ a2 u 2 ω )du2 +2 abuv ω dudv v 2 +(1+b2 ω )dv2, II = ωγ ( 1/2 a(bv 2 + c)du 2 2abuvdudv + b(au 2 + c)dv 2) where γ = a(a +1)u 2 + b(b +1)v 2 + c.

4 2558 Dae Won Yoon and Ji Soon Jun From (2.4) the Laplacian operator Δ II of non-degenerate second fundamental form II of M 1 can be expressed as (3.2) Δ II = γ { 2abu abc u +2abv v + b( au 2 + c ) 2 u +2abuv 2 2 u v + a( bv 2 + c ) 2 }. v 2 From (3.1) and (3.2), the Laplacian Δ II x of x is given by Δ II x = 2 γ ( ) 2 (3.3) γ u, v, ω = c c x. We suppose that the surface M 1 satisfies Δ II x = λx. Ifλ 0, then, from (3.3) we have a = b = 1 and λ = 2 c. Thus, we have the following theorem. Theorem 3.1. If M 1 is a quadric surface of the first kind with non-degenerate second fundamental form in a Euclidean 3-space satisfying the equation Δ II x = λx for some non-zero constant λ, then, M 1 is an open part of an ordinary sphere. We assume that M 1 is a quadric surface of the first kind satisfying Δ II x = f(u)x for some non-zero smooth function f(u). Then we get a 1 and b = 1 so that f(u) = 2 a(a+1)u 2 +c. Thus, according to the signs of the c non-zero constants a and c we have the following theorem. Theorem 3.2. If M 1 is a quadric surface of the first kind with non-degenerate second fundamental form in a Euclidean 3-space satisfying the equation Δ II x = f(u)x for some non-zero smooth function f(u), then, M 1 is an open part of one of the following surfaces: 1. the ellipsoid given by p 2 x 2 + y 2 + z 2 = r 2, 2. the hyperboloid of one sheet given by p 2 x 2 + y 2 + z 2 = r 2, 3. the hyperboloid of two sheet given by p 2 x 2 y 2 z 2 = r 2. Let G be the Gauss map of the surface M 1 in a Euclidean 3-space. Then the Gauss map G is obtained by (3.4) G = γ 1/2 ( au, bv, ω ). From (3.2) and (3.4), we get (3.5) where Δ II G = γ 2 (aur 1,bvR 2, ωr 3 ), R 1 =(3a + b +2)c + a(a + 1)(b +2)u 2 + b(b + 1)(3a 2b +2)v 2, R 2 =(a +3b +2)c a(a + 1)(2a 3b 2)u 2 +(a +2)b(b +1)v 2, R 3 =(a + b)c a(a + 1)(2a b)u 2 + b(b + 1)(a 2b)v 2.

5 Non-degenerate quadric surfaces 2559 We suppose that the surface M 1 satisfies Δ II G = λg with λ 0. Then, from (3.4) and (3.5) we have a = b = 1 and c = 4. Thus, we have the following λ 2 theorem. Theorem 3.3. Let M 1 be a quadric surface of the first kind with nondegenerate second fundamental form in a Euclidean 3-space. If the Gauss map G of M 1 satisfies Δ II G = λg for some non-zero constant λ, then, M 1 is an open part of an ordinary sphere. Secondly, we consider a quadric surface of the second kind M 2 in E 3. Then M 2 is parametrized by (3.6) x(u, v) =(u, v, a 2 u2 + b 2 v2 ). In this case, the first and the second fundamental form are given by, respectively, I =(1+a 2 u 2 )du 2 +2abuvdudv +(1+b 2 v 2 )dv 2,II= q 1/2 (adu 2 + bdv 2 ) where q =1+a 2 u 2 + b 2 v 2. On the other hand, the Gauss map G of the surface M 2 is given by G = q 1/2 ( au, bv, 1). By using (2.4), we can give the formula of the Laplacian Δ II of M 2 as follows ( Δ II = q a u + 1 ) 2 (3.7). 2 b v 2 By a straightforward computation, the Laplacian Δ II x and Δ II G of x and G can be expressed as, respectively Δ II x =(0, 0, 2 q), Δ II G = q 2( aur 1,bvR 2,R 3 ), where R 1 =3a + b + a 2 bu 2 + b 2 (3a 2b)v 2, R 2 = a +3b a 2 (2a 3b)u 2 + ab 2 v 2, R 3 = a b + a 2 (2a b)u 2 b 2 (a 2b)v 2. Thus, we have the following theorems. Theorem 3.4. There is no a quadric surface of the second kind with nondegenerate second fundamental form in a Euclidean 3-space satisfying Δ II x = λx. Theorem 3.5. There is no a quadric surface of the second kind with nondegenerate second fundamental form in a Euclidean 3-space satisfying Δ II G = λg. 4. Laplacian of the second fundamental form of ruled surfaces It is well known that a cylindrical ruled surface is flat, i.e., one of the principal curvatures is zero. Thus, non-cylindrical ruled surfaces are meaningful for our study.

6 2560 Dae Won Yoon and Ji Soon Jun Let M be a non-cylindrical ruled surface in E 3. Then the parametrization for M is given by (4.1) x = x(u, v) =α(u)+vβ(u) such that β,β =1, β,β = 1 and α,β = 0. In this case α is the striction curve of x, and the parameter u is the arc-length on the spherical curve β. Then, the first fundamental form of M is given by I =( α,α + v 2 )du 2 +2 α,β dudv + dv 2. For later use, we define the smooth functions Q, J and D as follows: Q = α,β β 0, J = β,β β, D = g 11 g 22 g12 2. In terms of the orthonormal basis {β,β,β β } we obtain (4.2) α = g 12 β + Qβ β,β = β Jβ β,α β = Qβ,β β = Jβ, which imply we have g 11 g 22 g12 2 = Q 2 + v 2. By using (4.2), the second fundamental form is given by (4.3) II = D 1 (Q(g 12 + QJ) Q v + Jv 2 )du 2 +2QD 1 dudv. From (2.4) and (4.3) the Laplacian Δ II of M can be expressed as follows: (4.4) Δ II = D Q {2 2 u v + 1 ( Q 2Jv ) Q v 1 ( Q(g12 + QJ) Q v + Jv 2) 2 }. Q v 2 From (4.1) and (4.4), the Laplacian Δ II x of x is obtained by Δ II x = D ( 2β + 1 Q Q Q( 2Jv ) ) (4.5) β. Suppose that the surface M satisfies Δ II x = λx. Then from (4.5) we have ( D 2β Q + ( Q 2Jv ) ) (4.6) β = λ(α + vβ)q. If λ = 0, that is, M satisfies the condition Δ II x = 0, then, from (4.6) we can easily obtain D = 0. It is a contradiction. If λ 0, then, by taking the inner product of β into (4.6), we have D ( Q 2Jv ) = λ( α, β + v)q 2. From this, we get the polynomial equation with respect to v which is (4.7) (Q 2 + v 2 ) ( Q 2Jv )2 = λ 2 Q 4 ( α, β 2 +2 α, β v + v 2 ). By comparing the coefficient of v 4, we have J = 0, so that α, β =0by comparing the coefficient of v. These induce (Q 2 + v 2 )(Q ) 2 = λ 2 Q 4 v 2. But, the last equation contradicts to λ 0,Q 0. Thus, we have the following theorem.

7 Non-degenerate quadric surfaces 2561 Theorem 4.1. There is no ruled surface with non-degenerate second fundamental form in a Euclidean 3-space satisfying Δ II x = λx. Let M be the ruled surface defined by (4.1). Then the Gauss map G of M is given by (4.8) G = D 1 (Qβ vβ β ). By a long computation, the Laplacian Δ II G of G can be expressed as (4.9) Δ II G =2vD 2 β + Q 1 D 4 {6Q 2 Q v +(Q 2 2v 2 )(Jv 2 Q v + Qg 12 + Q 2 J) + D 2 (2Jv 2 3Q v 2Q 2 J)}β + D 4 {3Q v 2 + D 2 (4Jv 5Q ) 3Jv 3 (3Qg 12 +3Q 2 J)v +6Q 2 Q }β β. Suppose that the surface M satisfies Δ II G = λg. By using (4.8) and (4.9) and by taking the inner product of β into the equation, we have 2v =0, it is a D 2 contradiction. Theorem 4.2. There is no ruled surface with non-degenerate second fundamental form in a Euclidean 3-space satisfying Δ II G = λg. Combining Theorem 3.1, Theorem 3.3, Theorem 3.4, Theorem 3.5, Theorem 4.1 and Theorem 4.2, we obtain the following Theorem 4.3. If M is a quadric surface with non-degenerate second fundamental form in a Euclidean 3-space satisfying the equation Δ II x = λx for some non-zero constant λ, then, M is an open part of an ordinary sphere. Theorem 4.4. If M is a quadric surface with non-degenerate second fundamental form in a Euclidean 3-space satisfying the equation Δ II G = λg for some non-zero constant λ, then, M is an open part of an ordinary sphere. Combining Theorem 4.3, Theorem 4.4 and the result of [ 9, 5 ], we have Theorem 4.5 (Characterization). Let M be a quadric surface with nondegenerate second fundamental form in a Euclidean 3-space. Then for some non-zero constant λ the following are equivalent: 1. Δx = λx. 2. Δ II x = λx. 3. M is an open part of an ordinary sphere. Theorem 4.6 (Characterization). Let M be a quadric surface with nondegenerate second fundamental form in a Euclidean 3-space. Then for some non-zero constant λ the following are equivalent: 1. ΔG = λg. 2. Δ II G = λg. 3. M is an open part of an ordinary sphere.

8 2562 Dae Won Yoon and Ji Soon Jun Acknowledgments The first author was supported by the fund of Research Promotion Program(RPP ), Gyeongsang National University. References [1] B.-Y. Chen and F. Dillen, Quadrics of finite type, J. Geom. 38 (1990), [2] B.-Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc. 44 (1987), [3] F. Dillen, J. Pas and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Insto. Math. Acad. Sinica 18 (1990), [4] O. J. Garay, An extension of Takahashi s theorem, Geom. Dedicate 34 (1990), [5] C. Jang, Surfaces with 1-type Gauss map, Kodai Math. J. 19 (1996), [6] G. Kaimakamis and B. Papantoniou, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying Δ II r = A r, J. Geom. 81 (2004), [7] Y. H. Kim, C. W. Lee and D. W. Yoon, On the Gauss map of surfaces of revolution without parabolic points, Bull. Korean Math. Soc. 46 (2009), [8] E. A. Ruh and J. Vilms, The tension field of the Gauss map, Trans. Amer. Math. Soc. 149 (1970), [9] T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), [10] D. W. Yoon and M. H. Kim Weingarten quadric surfaces in a Euclidean 3-space, Turk J. Math. 35 (2011), Received: May, 2012