CLASSIFICATION OF CYLINDRICAL RULED SURFACES SATISFYING H = AH IN A 3-DIMENSIONAL MINKOWSKI SPACE

Transcription

1 Bull. Korean Math. Soc. 33 (1996), No. 1, pp CLASSIFICATION OF CYLINDRICAL RULED SURFACES SATISFYING H = AH IN A 3-DIMENSIONAL MINKOWSKI SPACE SEOUNG DAL JUNG AND JIN SUK PAK ABSTRACT. 1. Introduction The study of surfaces in a Euclidean space whose Gauss map G satisfies the condition : G = AG ( ) for some matrix A was studied by C. Baikoussis, D. E. Blair, B. Y. Chen, F. Dillen, L. Verstraelen ([1], [2], [7]) and so on. Also, S. M. Choi ([6]) extended this problem to the Minkowski space and obtained the following theorem : THEOREM A. The only space-like or time-like ruled surfaces in R 3 1 whose Gauss map G : M M 2 (ɛ) satisfies the condition ( ) are locally the following spaces ; (1) The Minkowski plane R 2 1, the Lorentz hyperbolic cylinder S1 1 R and the Lorentz circular cylinder R 1 1 S1 if ɛ = 1, i.e., M 2 (1) := S 2 1 (1), (2) the Euclidean plane R 2 and the hyperbolic cylinder H 1 R if ɛ = 1, i.e., M 2 ( 1) := H 2 ( 1). Also, in 1994, B. Y. Chen ([5]) studied the submanifolds of Euclidean spaces satisfying H = AH ( ), where H is the mean curvature vector. This condition ( ) is a generalization of the condition ( ). In fact, the examples appeared in Theorem A satisfy ( ). Received August 22, AMS Subject Classification: 53A10, 53C50. Key words and phrases: Cylindrical ruled surface, Mean curvature, Space-like and time-like ruled surface. The Present Studies were Supported (in part) by the Basic Science Research Institute program, Ministry of Education, 1995, Project No, BSRI and TGRC-KOSEF

2 Seoung Dal Jung and Jin Suk Pak In this paper, we extend Theorem A under the condition ( ) and prove the following theorem : THEOREM. The only space-like or time-like cylindrical ruled surfaces in R 3 1 whose mean curvature vector H satisfies the condition ( ) are (1) and (2) in Theorem A. Throughout this paper, we assume that all objects are smooth and all surfaces are connected, unless otherwise mentioned. 2. Preliminaries Let R 3 1 given by be the 3-dimensional Minkowski spaces with the standard metric (2.1) g = dx 2 0 +dx2 1 +dx2 2, where (x 0, x 1, x 2 ) is a rectangular system of R 3 1. Let I and J be open intervals containing 0 in R. Letα=α(u) be a curve on J into R 3 1 and β = β(u) a vector field along α orthogonal to α. A ruled surface M in R 3 1 is defined as a semi-riemannian surface swept out by the vector field β along the curve α. ThenMalways has a parametrization (2.2) x(u,v)=α(u) + vβ(u), u J, v I, where we call α abasecurveandβa director curve. In particular, if β is constant, then it is said to be cylindrical, and if it is not so, then the surface is said to be non-cylindrical. The natural basis {x u, x v } along the coordinate curves are given by x u = dx( u ) = α +vβ, Accordingly it following that x v = dx( v ) = β. (2.3) g(x u, x u ) = g(α,α )+2vg(α,β )+v 2 g(β,β ), g(x u, x v ) = 0, g(x v, x v ) = g(β, β). 98

3 Classification of cylindrical ruled surfaces satisfying H = AH Since M is a semi-riemannian surface, it suffices to consider the cases that α is a space-like or time-like curve and β is a unit space-like or time-like vector field. The ruled surface M is said to be of type I or type II, according as the base curve α is space-like or time-like. First, we devide the ruled surface of type I into three types. In the case that β is space-like, it is said to be type I 0 + or I +, according as β is null or non-null. Since we have g(β, β ) = 0, when β is time-like, β is to be space-like. Hence we call this type as type I.On the other hand, for the ruled surface of type II, it is also said to be of type I I 0 + or II +, according as β is null or β is non-null. Notice that in case of type II the director curve β always is space-like. Then the ruled surface of type I + or I 0 + (resp. I, II + or II 0 + ) is space-like (resp. time-like). Denoting (g ij ) (resp. G) the inverse matrix (resp. the determinant) of the matrix (g ij ). Then the Laplacian on M is given by (2.4) = 1 ( G g ij ), G u j u i where u 1 = u and u 2 = v. LetNbe a unit normal vector to M. Itisdefined by f 1 x u x v, where f is the norm of the vector x u x v. Then the mean curvature vector H is defined by (2.5) H = 1 2 Gl + En 2Fm EG F 2 N, where E = g(x u, x u ), F = g(x u, x v ), G = g(x v, x v ), l = g(n, x uu ), m = g(n, x uv ) and n = g(n, x vv ). 3. Cylindrical ruled surfaces Let M be a cylindrical ruled surface of type I +, II + parametrized by x = x(u,v)=α(u) + vβ, where β is a unit space-like constant vector along the curve α orthogonal to it. That is, it satisfies g(α,β) = 0, g(β, β) = 1. Acting a Lorentz transformation, we may assume that β = (0, 0, 1) without loss of generality. Then α 99

4 Seoung Dal Jung and Jin Suk Pak may be regarded as the plane curve α(u) = (α 0 (u), α 1 (u), 0) parametrized by arc-length ; g(α,α )= α 02 +α 12 = ɛ. From (2.5), the mean curvature vector is given by H = ɛ 2 (α 0,α 1, 0), because of x uu orthogonal to x u and x v. It is the space-like or time-like vector to M, according as ɛ = 1or 1. Since the induced semi-riemannian metric g is given by g 11 = ɛ, g 12 = 0and g 22 = 1, the Laplacian of H is given by H = 1 2 (α(4) 0,α(4) 1,0)from (2.4). Thus, from the condition ( ) we have the following system of differential equations : ɛ 0 = a 11 α 0 + a 12α 1, (3.1) ɛ 1 = a 21 α 0 + a 22α 1, 0 = a 31 α 0 + a 32α 1, where A = (a ij )is the constant matrix. To solve this equation, condider that M is of type I +, i.e., the plane curve α is space-like (ɛ = 1). So we get g(α,α )= α 0 2 +α 1 2 =1. Accordingly we can parametrize as follows : (3.2) α 0 = sinh θ, α 1 = cosh θ, where θ = θ(u). Differentiating (3.2), we obtain (3.3) α 0 = θ cosh θ, α 0 = θ cosh θ + θ 2 sinh θ, 0 = (θ + θ 3 ) cosh θ + 3θ θ sinh θ, α 1 = θ sinh θ, α 1 = θ sinh θ + θ 2 cosh θ, 1 = (θ + θ 3 ) sinh θ + 3θ θ cosh θ. By (3.1), (3.2) and (3.3), we have (θ + θ 3 ) cosh θ + 3θ θ sinh θ = a 11 θ cosh θ a 12 θ sinh θ, (θ + θ 3 ) sinh θ + 3θ θ cosh θ = a 21 θ cosh θ a 22 θ sinh θ, 100

5 Classification of cylindrical ruled surfaces satisfying H = AH which give (3.4) θ + θ 3 = a 11 θ cosh 2 θ θ (a 12 a 21 ) cosh θ sinh θ + a 22 θ sinh 2 θ (3.5) 3θ θ = a 21 θ cosh 2 θ + θ (a 11 a 22 ) cosh θ sinh θ + a 21 θ sinh 2 θ. Case i) θ 0. From (3.5), we have (3.6) 3θ = a 21 cosh 2 θ + (a 11 a 22 ) cosh sinh θ + a 12 sinh 2 θ. Differentiating (3.6), we get 3θ = θ (a 11 a 22 )(cosh 2 θ + sinh 2 θ)+2θ (a 12 a 21 ) cosh θ sinh θ. Substituting this equation into (3.4), we get (3.7) (a 11 a 22 )(cosh 2 θ + sinh 2 θ)+2(a 12 a 21 ) cosh θ sinh θ + 3θ 2 = 3{a 11 cosh 2 θ a 22 sinh 2 θ (a 21 a 12 ) sinh θ cosh θ}. Differentiating (3.7), we have (3.8) (5a 12 7a 21 ) cosh 2 θ + (7a 12 5a 21 ) sinh 2 θ + 12(a 11 a 22 ) cosh θ sinh θ = 0. From (3.1) and (3.8), we get (3.9) a 11 = a 22, a 12 = a 21 = a 31 = a 32 = 0, because sinh θ cosh θ,sinh 2 θ and cosh 2 θ are linearly independent functions of θ = θ(u). Combining (3.9) with (3.5), we have θ =± 1 r u+b, 101

6 Seoung Dal Jung and Jin Suk Pak where 1 r 2 = a 11 = a 22, r > 0, b R. Accordingly we have This representation gives us to α 0 =±rcosh θ + c 0, c 0 R, α 1 =±rsinh θ + c 1, c 1 R. (α 0 c 0 ) 2 + (α 1 c 1 ) 2 = r 2, r>0. We denote by H 1 (r,(c 0,c 1 )) the hyperbolic cicle centered at (c 0, c 1 ) with radius r in the Minkowski plane R 2 1. By the above equation the curve α is contained in H 1 (r,(c 0,c 1 )) and hence the ruled surface M is contained in the hyperbolic cylinder H 1 R. Case ii) θ = 0. Let J 0 beaset{u J θ (u)=0}. We claim that if J 0 is not empty, then J 0 is to be J itself. In fact, we suppose that J 0 J, i.e., J J 0 φ. Then (3.9) is satisfied on J J 0. Since A is constant matrix, (3.9)issatisfiedonJ. So (3.6) leads thst θ = 0onJ, i.e., θ is constant on J. By assumption, there exists u 0 J 0 and θ (u 0 ) = 0. Thus θ is zero on J, a contradiction. Hence θ is constant on J. Therefore the normal vector N is the time-like constant vector. This implies that M is contained in R 2. Next we are concerned with the cylindrical ruled surface M of type II +, i.e., the plane curve α is time-like (ɛ = 1). Then the surface M is time-like and we get g(α,α )= 1. Accordingly we can parametrize as follows : α 0 = cosh θ, α 1 = sinh θ, where θ = θ(u). By the similar discussion to that of the above ruled surface of type I + we can get, under θ 0, which yields that a 11 = a 22, a 12 = a 21 = a 31 = a 32 = 0, θ =± 1 r u+b, 1 r 2 =a 11 = a 22, r > 0, b R. 102

7 Classification of cylindrical ruled surfaces satisfying H = AH Accordingly we have α 0 =±rsinh θ + c 0, c 0 R, α 1 =±rcosh θ + c 1, c 1 R. This representation gives us to (α 0 c 0 ) 2 + (α 1 c 1 ) 2 = r 2, r > 0. We denote by S1 1(r,(c 0,c 1 )) the pseudo-circle centered at (c 0, c 1 ) with radius r in the Minkowski plane R 2 1. By the above equation the curve α is contained in S1 1(r,(c 0,c 1 )) and hence the ruled surface M is contained in the Lorentz circular cylinder S1 1 R. On the other hand, if a set {u J θ (u) = 0} is not empty, then θ is constant on J by the similar discussion to that about the surface of type I +. So we get that the normal vector N is the space-like constant vector. It shows that M is contained in R 2 1. Hence we have THEOREM 3.1. The only cylindrical ruled surfaces of type I + (resp. II + )in R 3 1whose the mean curvature vector satisfies the condition ( ) are locally the plane or the hyperbolic cylinder (resp. the Minkowski plane or the Lorentz circular cylinder). Now, let M be a cylindrical ruled surface of type I.ThenMis parametrized by x = x(u,v)=α(u) + vβ, where β is a unit time-like constant vector along the space-like curve α orthogonal to it. That is, it satisfies g(α,β) = 0, g(β, β) = 1. Acting a Lorentz transformation, we may assume that β = (1, 0, 0) without loss of generality. Then α is the plane curve α(u) = (0,α 1 (u), α 2 (u)) parametrized by arclength ; (3.10) g(α,α )=α 12 +α 22 =1. The Laplacian of H is given by H = 1 2 (0,α(4) 1,α(4) 2 ). Thus from the condition ( ), we have the following system of differential equations : 0 = a 12 α 1 a 13α 2, (3.11) 1 = a 22 α 1 a 23α 2, 2 = a 32 α 1 a 33α

8 Seoung Dal Jung and Jin Suk Pak From (3.10), we can parametrize as follows : (3.12) α 1 = cos θ, α 2 = sin θ, where θ = θ(u). Then, differentiating (3.12), we obtain (3.13) α 1 = θ sin θ, α 1 = θ sin θ θ 2 cos θ, 1 = ( θ + θ 3 ) sin θ 3θ θ cos θ, α 2 = θ cos θ, α 2 = θ cos θ θ 2 sin θ, 2 = (θ θ 3 ) cos θ 3θ θ sin θ. By (3.11), (3.12) and (3.13) we have Which give (3.14) ( θ + θ 3 ) sin θ 3θ θ cos θ = a 22 θ sin θ a 23 θ cos θ, (θ θ 3 ) cos θ 3θ θ sin θ = a 32 θ sin θ a 33 θ cos θ, θ θ 3 = a 33 θ cos 2 θ a 22 θ sin 2 θ + θ (a 32 + a 23 ) cos θ sin θ, (3.15) 3θ θ = a 23 θ cos 2 θ + a 32 θ sin 2 θ + θ (a 22 a 33 ) cos θ sin θ. Now, assume that Case i) θ 0. From (3.15), we have (3.16) 3θ = (a 22 a 33 ) cos θ sin θ a 23 cos 2 θ + a 32 sin 2 θ. Differentiating this equation, we get (3.17) 3θ = θ (a 22 a 33 )(cos 2 θ sin 2 θ)+2θ (a 23 + a 32 ) cos θ sin θ. Substituting (3.17) into (3.14), we get (3.18) (a 22 4a 33 ) cos 2 θ + (a 33 4a 22 ) sin 2 θ + 5(a 23 + a 32 ) cos θ sin θ + 3θ 2 =

9 Classification of cylindrical ruled surfaces satisfying H = AH Differentiating (3.18), we get (3.19) 6θ = 10(a 22 a 33 ) cos θ sin θ 5(a 23 + a 32 )(cos θ sin 2 θ). From (3.16) and (3.19), we have (3.20) (5a a 32 ) sin 2 θ (7a a 32 ) cos 2 θ + 12(a 22 a 33 ) cos θ sin θ = 0. Hence from (3.20) and (3.11), a 12 = a 13 = a 23 = a 32 = 0, a 22 = a 33, which yields that θ =± 1 r u+b, 1 r 2 =a 22 = a 33, r > 0, b R. Accordingly, we have This representation gives us to α 1 =±rsin θ + c 1, c 1 R, α 2 = rcos θ + c 2, c 2 R. (α 1 c 1 ) 2 + (α 2 c 2 ) 2 = r 2, r > 0. We denote by S 1 (r,(c 1,c 2 )) the circle centered at (c 1, c 2 ) with radius r in the plane R 2. By the above equation the curve α is contained in S 1 (r,(c 1,c 2 )) and hence the ruled surface M is contained in the Lorentz circular cylinder R 1 1 S1. Case ii) θ = 0. By similar calculation with in Theorem 3.1, θ is constant on J. So we get that the normal vector N is the space-like constant vector. It shows that M is contained in R 2 1. Thus we have THEOREM 3.2. The only cylindrical ruled surfaces of type I in R 3 1 whose the mean curvature vector H satisfies ( ) are locally the Minkowski plane or the circular cylinder of index

10 Seoung Dal Jung and Jin Suk Pak References 1. C. Baikoussis and D. E. Blair, On the Gauss map of ruled surface, Glasgow Math. J. 34 (1992), C. Baikoussis, B. Y. Chen and L. Verstraelen, Ruled surfaces and tubes with finite type Gauss map, Tokyo J. Math. 16 (1993), B. Y. Chen, Some classification theorems for submanifolds in Minkowski space- time, Archiv Math. 62 (1994), B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), B. Y. Chen, Submanifolds of Euclidean spaces satisfying H = AH, Tamkang J. Math 25 (1994), S. M. Choi, On the Gauss map of ruled surfaces in a 3-dimensional Minkowski space, to appear in Tsukuba J. Math. (to appear). 7. F. Dillen, J. Pas and L. Verstraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica 18 (1990), SEOUNG DAL JUNG, DEPARTMENT OF MATHEMATICS,CHEJU NATIONAL UNIVERSITY, CHEJU , KOREA JIN SUK PAK, DEPARTMENT OF MATHEMATICS,KYUNGPOOK NATIONAL UNIVERSITY, TAEGU , KOREA 106