On Null 2-Type Submanifolds of the Pseudo Euclidean Space E 5 t

Transcription

1 International Mathematical Forum, 3, 2008, no. 3, On Null 2-Type Submanifolds of the Pseudo Euclidean Space E 5 t Güler Gürpınar Arsan, Elif Özkara Canfes and Uǧur Dursun Istanbul Technical University, Faculty of Science and Letters Department of Mathematics, Maslak, Istanbul, Turkey ggarsan@itu.edu.tr, canfes@itu.edu.tr, udursun@itu.edu.tr Abstract We mainly prove that a space-like submanifold M in the pseudo- Euclidean space Et 5, t =, 2, with constant mean curvature and nonparallel mean curvature vector is flat and of null 2-type if and only if M is an open portion of a 3-dimensional helical cylinder of the first kind or a 3-dimensional helical cylinder of the second kind in Et 5,t=, 2. Mathematics Subject Classification: Primary 53C40 Keywords: Finite type submanifold, null 2-type submanifold, mean curvature vector, helical cylinder Introduction A connected submanifold M n of a pseudo-euclidean space Et m is called of finite type if its position vector field x can be written as a sum of eigenfunctions of its Laplacian; more precisely, M n is said to be of finite k-type if its position vector field x admits the following spectral decomposition x = x 0 + x + + x k, ) where Δx i = λ i x i,i=, 2,...,k, λ < <λ k,x 0 is a constant vector in E m t and x,...,x k are non-constant E m t -valued maps on M n. If one of the eigenvalues λ i vanishes, then M n is said to be of null k-type see [, 2] for detail). We can choose a coordinate system on E m t with x 0 as its origin. Then we have the following simple spectral decomposition of x for a null 2-type submanifold M: x = x + x 2, Δx =0, Δx 2 = λx 2. 2)

2 60 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun In [4, 5], B.Y. Chen gave a classification of null 2-type surfaces in the Euclidean space E 3 and E 4. He proved that circular cylinders and helical cylinders are the only surfaces of null 2-type in E 3 and E 4, respectively. In [5], he also proved that a surface M in the Euclidean space E 4 is of null 2-type with parallel normalized mean curvature vector if and only if M is an open portion of a circular cylinder in a hyperplane of E 4. However, in [3], S.J. LI showed that a surface M in E m with parallel normalized mean curvature vector is of null 2-type if and only if M is an open portion of a circular cylinder. Later, in [6], B.Y. Chen and H. Song proved that a space-like surface M in Et 4,t=, 2, is of null 2-type with constant mean curvature if and only if M is an open portion of a helical cylinder of the first kind or a helical cylinder of the second kind in Et 4,t=, 2. Also, in [2], D.S. Kim and Y.H. Kim gave complete classification theorems on null 2-type surfaces in Minkowski space E 4. They proved that a Lorentzian surface M in E 4 is of null 2-type with constant mean curvature if and only if M is an open portion of a helical cylinder of third kind, a helical cylinder of fourth kind, an extended B-scroll or a cylinder E S r), S r) E. In the case of the classification of hypersurfaces, the constancy of the mean curvature does not provide enough information to obtain a characterization of null 2-type hypersurfaces of Euclidean spaces and Lorentzian spaces. In [0, ], A. Ferrandez and P. Lucas studied null 2-type hypersurfaces of Euclidean spaces and null 2-type space-like hypersurfaces of Lorentzian spaces with additional assumption of having at most two distinct principal curvatures. They proved that Euclidean hypersurfaces of null 2-type and having at most two distinct principal curvatures are locally isometric to a generalized spherical cylinder, [0], and a space-like hypersurface of the Lorentzian space E m with at most two distinct principal curvatures is of null 2-type if and only if it is locally isometric to a generalized hyperbolic cylinder, []. The assumptions on a hypersurface to be of null 2-type are not enough for a submanifold M n,n 3, of an Euclidean space E m and a pseudo- Euclidean space Et m to be of null 2-type. In [7, 8, 9], the third author studied 3-dimensional null 2-type submanifolds of the Euclidean space E 5 and pseudo- Euclidean space Et 5,t=, 2. In [7], he proved that a 3-dimensional submanifold M of the Euclidean space E 5 having two distinct principal curvatures in the parallel mean curvature direction and having a second fundamental form of a constant square length is of null 2-type if and only if M is locally isometric to one of E S 2 E 4 E 5, E 2 S E 4 E 5 or E S a) S a). In [8], he prove that a 3-dimensional space-like submanifold M of the pseudo- Euclidean space Et 5 with parallel normalized non-null mean curvature vector is of null 2-type having two distinct principal curvatures in the mean curvature direction and having a constant scalar curvature if and only if M is locally isometric to one of the following:

3 On null 2-type submanifolds 6. S a) E 2 E 4 E 5 or S2 a) E E 4 E 5 when H is space-like, 2. H a) E 2 E 4 E 5 or H 2 a) E E 4 E 5 when H is time-like, or 3. H a) E 2 E 4 E 5 2, H 2 a) E E 4 E 5 2,or H a) H a) E E 5 2. On the other hand, in [9], he showed that a 3-dimensional submanifold M of the Euclidean space E 5 with constant mean curvature and non-parallel mean curvature vector is flat and of null 2-type if and only if M is an open portion of a 3-dimensional helical cylinder. Here we study null 2-type space-like submanifolds of Et 5,t=, 2 with nonparallel mean curvature direction. We firstly show that for a flat, null 2-type, space-like submanifold M 3 of the pseudo-euclidean space Et 5, t =, 2, the dimension of the first normal space N M) ofm is one if and only if M has constant mean curvature and non-parallel mean curvature vector. Then we prove that a 3-dimensional space-like submanifold M in the pseudo-euclidean space Et 5, t =, 2, is flat and of null 2-type with constant mean curvature and non-parallel mean curvature vector if and only if M is an open portion of a 3-dimensional helical cylinder of the first kind or a helical cylinder of the second kind in Et 5,t=, 2. 2 Preliminaries Let Et m by be an m-dimensional pseudo-euclidean space with metric tensor given g = t dx i ) 2 + i= dx i ) 2 i=t+ where x,...,x m ) is a rectangular coordinate system of Et m.soem t,g)isa flat pseudo-riemannian manifold with signature t, m t). When t =,E m is called the Lorentzian space. Let M be an n- dimensional pseudo-riemannian submanifold of an m- dimensional pseudo-euclidean space Et m. We denote by h, A, H, and, the second fundamental form, the Weingarten map, the mean curvature vector, the Riemannian connection and the normal connection of the submanifold M in Et m, respectively. Let e,...,e n,e n+,...,e m be an adapted local orthonormal frame in Et m such that <e A,e B >= ε B δ AB, ε B =<e B,e B >= ±), e,...,e n are tangent to Mt n and e n+,...,e m are normal to Mt n. We use the following convention on the range of indices: A,B,C,... m, i, j, k,... n, n + β,ν,γ,... m.

4 62 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun Let {ω A } be the dual -forms of {e A } defined by ω A X) =<e A,X >.The connection forms ωa B of Em t are defined by de A = ωa B e B, ε B ωa B + ε AωB A =0. B= For lifting or lowering indices we use ω A = ε A ω A, ωa B = ε Bω AB. Then the structure equations of Et m are obtained as follows dω A = ω B ωb, A dωa B = ωa C ωc B. 3) B= Restricting these forms to M we have ω β =0, dω β = C= ω i ω β i =0, β = n +,...,m. i= By Cartan s Lemma, we can write ω β i = h β ij ωj, h β ij = hβ ji, 4) j= where h β ij are coefficients of the second fundamental form in the direction e β. The mean curvature vector H is given by H = n The first equation of 3) gives β=n+ ε β trh β )e β. 5) dω i = ω j ωj i, ε iωj i + ε jω j i j= =0, 6) where {ωj i } is the connection forms on M and uniquely determined by these equations. Also, from the second equation of 3) we can have the Gauss and Codazzi equations, respectively, as m dω j i = ωi k ωj k + ω β i ωj β 7) k= β=n+ and dω β i = ωi k ωβ k + m k= ν=n+ ω ν i ωβ ν. 8)

5 On null 2-type submanifolds 63 Using 4) and the connection equations ei e j = ωj k e i)e k we can restate the equations of Gauss 7) and Codazzi 8) relative to the basis e,...,e n, respectively, as follows e l ω j i e k)) e k ω j i e l)) = k= {ωi r e l )ωre j k ) ωi r e k )ωre j l ) r= + ω j i e r)[ω r k e l) ω r l e k)]} + i<j n, l<k n, ν=n+ ε j ε ν ε k h ν ik hν jl ε lh ν jk hν il ), 9) and e j h ν ik ) e kh ν ij )= { h ν ir [ωr k e j) ωj r e k)] + h ν rk ωr i e j) h ν rj ωr i e k)} r= + h β ij ων β e k) h β ik ων β e j)), 0) β=n+ ν = n +,...,m, i=,...,n, j<k n. Also the Ricci equation is given by R e i,e j )e β,e γ = ε γ [A eβ,a eγ ]e i ),e j, ) where R is the curvature tensor of the normal bundle. The first normal space N M)ofM at each point p M in Et m is defined as the orthogonal complement of the subspace {ξ Tp M A ξ =0} in the normal space Tp M. 3 Examples of Null 2-type Submanifolds We need the following examples for later use. Example 3. Helical cylinder of first kind in E. 5 For any constants a, b with a 2 >b 2 we consider the following 3-dimensional space-like submanifold M in E 5 defined by bu xu,u 2,u 3 )= c,acos u c,asin u ) c,u 2,u 3, with c = a 2 b 2 which is called a 3-dimensional helical cylinder of the first kind in E 5. The metric tensor g of M is given by g = du2 + du2 2 + du2 3 and thus the submanifold M is flat.

6 64 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun If we put x = ) bu c, 0, 0,u 2,u 3 and x 2 = 0,acos u c,asin u c, 0, 0 ) then we have Δx =0, Δx 2 = c 2 x 2. 2) This shows that M is a null 2-type and space-like submanifold in E 5. If we choose b e = c, a c sin u c, a c cos u ) c, 0, 0, e 2 =0, 0, 0,, 0), e 3 =0, 0, 0, 0, ) e 4 = 0, cos u c, sin u ) c, 0, 0, a e 5 = c, b c sin u c, b c cos u ) c, 0, 0, then we have ε = ε 2 = ε 3 = ε 4 =,ε 5 =. Moreover, the connection forms are given ω 2 = ω 3 = ω2 3 = ω4 2 = ω4 3 = ω5 = ω5 2 = ω5 3 =0, ω 4 = a c 2 ω, and the mean curvature vector H is given by H = a 3c e 2 4 ω 5 4 = b c 2 ω, ω i = du i,i=, 2, 3, 3) which is space-like and non-parallel. Example 3.2 Helical cylinder of second kind in E. 5 For any constants a, b 0 we consider the following 3-dimensional submanifold M in E 5 defined by xu,u 2,u 3 )= a cosh u c,asinh u c, bu ) c,u 2,u 3, with c = a 2 + b 2 which is called a 3-dimensional helical cylinder of the first second in E 5. Then M is a flat, null 2-type and space-like submanifold in E5. If we put a e = c sinh u c, a c cosh u c, b ) c, 0, 0, e 2 =0, 0, 0,, 0),

7 On null 2-type submanifolds 65 e 3 =0, 0, 0, 0, ) e 4 = cosh u c, sinh u ) c, 0, 0, 0, b e 5 = c sinh u c, b c cosh u ) c, a c, 0, 0, then we have ε = ε 2 = ε 3 = ε 5 =,ε 4 =. Moreover, the connection forms are given ω 2 = ω 3 = ω 2 3 = ω 4 2 = ω 4 3 = ω 5 = ω 5 2 = ω 5 3 =0, ω 4 = a c 2 ω, ω 5 4 = b c 2 ω, ω i = du i,i=, 2, 3, 4) and the mean curvature vector H is given by H = a 3c 2 e 4 which is time-like and non-parallel. Example 3.3 Helical cylinder of first kind in E2 5. For any constants a, b with a 2 >b 2 we consider the 3-dimensional submanifold M in E2 5 defined by xu,u 2,u 3 )= b sin u c,bcos u c, au ) c,u 2,u 3, with c = a 2 b 2 which is called a 3-dimensional helical cylinder of the first kind in E2 5. Then M is a flat, null 2-type and space-like submanifold in E5 2. If we put b e = c cos u c, b c sin u c, a ) c, 0, 0, e 2 =0, 0, 0,, 0), e 3 =0, 0, 0, 0, ) e 4 = sin u c, cos u ) c, 0, 0, 0, a e 5 = c cos u c, a c sin u c, b ) c, 0, 0, then we have ε = ε 2 = ε 3 =,ε 4 = ε 5 =. Moreover, the connection forms are given ω 2 = ω 3 = ω2 3 = ω4 2 = ω4 3 = ω5 = ω5 2 = ω5 3 =0, ω 4 = b c 2 ω, ω 5 4 = a c 2 ω, ω i = du i,i=, 2, 3, 5) and the mean curvature vector H is given by H = b 3c 2 e 4

8 66 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun which is timelike and non-parallel. Example 3.4 Helical cylinder of second kind in E2 5 For any constants a, b with a 2 >b 2 we consider the 3-dimensional submanifold M in E2 5 defined by bu xu,u 2,u 3 )= c,acosh u c,asinh u ) c,u 2,u 3, with c = a 2 b 2 which is called a helical cylinder of the second kind in E2. 5 Then M is a flat, null 2-type and space-like submanifold in E2 5. If we put b e = c, a c sinh u c, a c cosh u ) c, 0, 0, e 2 =0, 0, 0,, 0), e 3 =0, 0, 0, 0, ), e 4 = 0, cosh u c, sinh u ) c, 0, 0, a e 5 = c, b c sinh u c, b c cosh u ) c, 0, 0, then we have ε = ε 2 = ε 3 =, ε 4 = ε 5 =. Moreover, the connection forms are given ω 2 = ω 3 = ω2 3 = ω4 2 = ω4 3 = ω5 = ω5 2 = ω5 3 =0, ω 4 = a c 2 ω, and the mean curvature vector H is given by which is time-like and non-parallel. ω 5 4 = b c 2 ω, ω i = du i,i=, 2, 3, H = a 3c 2 e 4 6) 4 Null 2-type space-like submanifolds of E 5 t In [3], -type pseudo-riemannian submanifolds of a pseudo-euclidean space Et m were completely classified. They are minimal submanifolds of Et m, minimal submanifolds of a pseudo-riemannian sphere in Et m or minimal submanifolds of a pseudo-hyperbolic space in Et m. For a null 2-type submanifold M of Et m, using Δx = nh the definition 2) implies ΔH = λh. 7)

9 On null 2-type submanifolds 67 Lemma 4. []) Let M be an n-dimensional pseudo-riemannian submanifold of a pseudo-euclidean space Et m. Then, there is a constant λ 0 such that ΔH = λh holds if and only if M is either of -type or of null 2-type. If the mean curvature vector H is non-null, that is, H, H 0, then there is an orthonormal normal frame e n+,...,e m such that H = αe n+, where α 2 = ε n+ H, H. For a null 2-type submanifold we can have Lemma 4.2 [6]) Let M be an n-dimensional pseudo-riemannian submanifold of E m t. If M is not of -type and H = αe n+ is non-null, then M is of null 2-type if and only if we have n 2 ε n+ α 2 )+2A n+ α)+2α Δα = λα αε n+ A n+ 2 α i= ν=n+2 i= ν=n+2 ε β tra H A β )=2ω β n+ α)+α tr ω β n+)+α ε i ω ν n+e i )A eν e i )=0. 8) ε i ε ν ε n+ ω ν n+e i )) 2, 9) i= ν=n+2 ε i ω ν n+e i )ω β ν e i ), 20) where β = n +2,...,m, A n+ 2 = n i= ε i <A n+ e i,a n+ e i >, tr ω β n+ )= n i= ε i ei ω β n+ )e i) and λ is a nonzero constant. The details of the characterization of finite type submanifolds can be seen in [, 2, 3, 5]. Let M be a 3-dimensional space-like submanifold of E 5 t and choose e,e 2,e 3 in such a way that e,e 2,e 3 diagonalize A e4. If M is flat, then we can have some of the equations of Gauss and Codazzi we need as follows: the equations of Gauss from 9) are h 5 3h 5 2 = h 5 23h 5, 2) h 5 23 h5 2 = h5 3 h5 22, 22) h 5 3 h5 23 = h5 2 h5 33, 23) ε 4 h 4 h ε 5 h 5 h 5 22 h 5 2) 2 )=0, 24)

10 68 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun ε 4 h 4 h ε 5h 5 h5 33 h5 3 )2 )=0, 25) ε 4 h 4 22h ε 5 h 5 22h 5 33 h 5 23) 2 )=0, 26) the equation of Codazzi from 0) for ν =4is h 5 23ω4e 5 )=h 5 3ω4e 5 2 ), 27) and the equations of Codazzi from 0) for ν = 5 are e h 5 2 ) e 2h 5 )=h4 ω5 4 e 2), 28) e h 5 3) e 3 h 5 ) =h 4 ω 5 4e 3 ), 29) e h 5 22 ) e 2h 5 2 )= h4 22 ω5 4 e ), 30) e h 5 33 ) e 3h 5 3 )= h4 33 ω5 4 e ). 3) Theorem 4.3 Let M be a 3-dimensional, flat, null 2-type and space-like submanifold of the pseudo-euclidean space Et 5, t =, 2, such that M does not lies in a hyperplane of Et 5. Then, the dimension of the first normal space N M) of M is one if and only if M has constant mean curvature and nonparallel mean curvature vector. Proof: Assume that dim N M)) =. Then, for the orthonormal normal basis e 4 = H α,e 5 we have A 5 =0, i.e., h 5 ij =0,i,j=, 2, 3. Hence, it follows from the equation of Ricci ) that the normal space is flat. Since M is of null 2-type then the equation 8) implies A 4 α) = 3 2 αε 4 α, that is, α is an eigenvector of A 4 with the eigenvalue 3αε 2 4 on U = {p M : α 0 at p}. If we choose e parallel to α, then h 4 = 3αε 4 2, h h 4 33 = 9αε 4 and e 2 α) =e 3 α) =0. As M is flat, i.e., ωj i 0, then the 2 equations of Codazzi 0) for ν = 4 give e j h 4 ik ) e kh 4 ij )=0, i =, 2, 3, j<k 3. For j =,k= i = 2 and j =,k= i = 3 we get e h 4 22 ) = 0 and e h 4 33 )=0, respectively. Therefore we obtain e h h 4 33) = 9 2 ε 4e α) = 0 which implies

11 On null 2-type submanifolds 69 that, α =0onU, i.e., U =, and α is a constant. As A 5 = 0, from the Weingarten formula we have De 5 = e 5, where D denotes the Riemannian connection of Et m. If the vector e 5 is parallel, then it is constant and M lies in a hyperplane of Et 5 which is not possible because of the hypothesis. Therefore e 5 is non-parallel and as the codimension is two, e 4 is also non-parallel, that is, the mean curvature vector H = αe 4 is non-parallel. Conversely, suppose that the mean curvature α is constant and the mean curvature vector H is non-parallel. As M is of null 2-type, then α 0, and thus, each normal vector in the orthonormal normal frame {e 4 = H,e α 5} is non-parallel and also tra 5 = h 5 + h h5 33 =0. Now, from the first two equations of Lemma 4.2 we have h 5 ω5 4 e )+h 5 2 ω5 4 e 2)+h 5 3 ω5 4 e 3)=0, 32) h 5 2 ω5 4 e )+h 5 22 ω5 4 e 2)+h 5 23 ω5 4 e 3)=0, 33) h 5 3 ω5 4 e )+h 5 23 ω5 4 e 2)+h 5 33 ω5 4 e 3)=0, 34) ε 4 ε 5 [ω 5 4 e )) 2 +ω 5 4 e 2)) 2 +ω 5 4 e 3)) 2 ]+ε 4 [h 4 )2 +h 4 22 )2 +h 4 33 )2 ]=λ. 35) Since e 4 is non-parallel in the normal space then ω4 5 0, that is, ω5 4 e i) 0 for some i {, 2, 3}. Without losing generality suppose that ω4e 5 ) 0. The equations 32)-34) form a system of linear equations with respect to ω4 5e ),ω4 5e 2) and ω4 5e 3). Since ω4 5e ) 0 we can have deth 5 ij ) = 0, that is, deth 5 ij) =h 5 h 5 22h 5 33 h 5 23) 2 )+h 5 2h 5 3h 5 23 h 5 2h 5 33) 36) + h 5 3h 5 2h 5 23 h 5 3h 5 22) =0. Using 22) and 23) we have deth 5 ij ) = h5 h5 22 h5 33 h5 23 )2 ) = 0. Hence h 5 22h 5 33 =h 5 23) 2 or h 5 =0. For each case we will show that A 5 =0. Case I. h 5 22 h5 33 =h5 23 )2. From 26) we get ε 4 h 4 22 h4 33 =0. If h4 22 =0, then the equation 24) implies h 5 h 5 22 =h 5 2) 2. Replacing h 5 = h 5 22 h 5 33 into the equation h 5 h5 22 = h5 2 )2 and using the equation h 5 22 h5 33 = h5 23 )2 we obtain h 5 2 )2 +h 5 23 )2 +h 5 22 )2 =0, that is, h 5 2 = h5 23 = h5 22 =0. Also, considering ω4e 5 ) 0 and using h 5 33 = h 5 the equations 32) and 34) imply h 5 = h5 3 =0. Hence we obtain A 5 =0. Ifh 4 33 =0, then we can similarly get A 5 = 0 by using 25), the equations 32) and 33). Case II. h 5 =0. Then h5 22 = h5 33 and from 2) we get h5 3 h5 2 =0. We will show that if one of h 5 3,h5 2 is zero then the other must be zero. Suppose that h and h 5 3 =0. Then, by 27) we have h 5 23 =0 as ω4e 5 ) 0,

12 620 Güler Gürpınar Arsan, Elif Özkara Canfes, Uǧur Dursun and hence, by 23) we get h 5 33 =0, i.e., h5 22 = 0. Therefore the equation 33) yields h 5 2 = 0 which is a contradiction. Similarly, it can be shown that h5 3 =0 when h 5 2 =0. Now,ash 5 2 = h 5 3 = 0, the equation 27) implies that h 5 23 =0 because of ω4 5e ) 0. By using h h5 33 = 0, the sum of the equations 30) and 3) implies that h h4 33 )ω5 4 e )=0, that is, h 4 33 = h4 22 and then h 4 =3αε 4 0. Hence the equation 26) gives ε 4 h 4 22) 2 + ε 5 h 5 22) 2 =0. Now, if t = 2, i.e., ε 4 ε 5 =, then we obtain h 5 22 = h4 22 = 0. So it follows that A 5 =0. Ift =, i.e., ε 4 ε 5 =, then we have h 4 22) 2 =h 5 22) 2. From 24) we get ε 4 h 4 h4 22 = 0 which gives h4 22 =0ash4 =3αε 4 0, and thus h 5 22 =0. Therefore A 5 =0. As a results, dim N M)) = because of A 4 0. Theorem 4.4 Let M be a 3-dimensional space-like submanifold in the pseudo-euclidean space Et 5,t=, 2. Then M is a flat, null 2-type and spacelike submanifold in the pseudo-euclidean space Et 5, t =, 2, with constant mean curvature and non-parallel mean curvature vector if and only if M is an open portion of a 3-dimensional helical cylinder of the first kind or a helical cylinder of the second kind in Et 5,t=, 2. Proof: Let M be an open portion of a 3-dimensional helical cylinder of the first kind or the second kind in Et 5, t =, 2, then M is a flat, null 2-type space-like submanifold of Et 5 with constant mean curvature and non-parallel mean curvature vector, cf. Examples ). Conversely, let M be a flat, null 2-type and space-like submanifold of Et 5 with constant mean curvature and non-parallel mean curvature vector. Then, we have dim N M)) = by Theorem 4.3. This means that A 5 =0, i.e., h 5 ij =0, i,j =, 2, 3. Since e 4 = H is not parallel in the normal space, then α ω4 5 0, that is, ω5 4 e i) 0 for some i {, 2, 3}. As in the proof of Theorem 4.3, without losing generality, we can take ω4 5e ) 0. Hence, using 30) and 3) we get h 4 22 = h 4 33 =0, and so h 4 =3αε 4 0. Moreover we have ω4 5e 2)=ω4 5e 3) = 0 from the equations 28) and 29). As M is null 2-type, then the equation 35) implies that μ = ω4 5e ) is a constant. Therefore we obtain ω 4 2 = ω4 3 = ω5 = ω5 2 = ω5 3 =0, ω4 =3αε 4 ω, ω 5 4 = μω. 37) Considering the flatness of M and 37), we conclude that M is an open portion of the product of a space-like 2-plane and a space-like 2-type curve in Et 3,say,σs), parameterized by arc length parameter with constant curvature κ =3α 0 and constant torsion τ = μ 0. In [6], it was shown that up to motions in Et 3,t =, 2, σs) is an open portion of one of the following space-like 2-type curve in Et 3 :ift = bs σs) = c,acos s c,asin s ), c

13 On null 2-type submanifolds 62 with c = a 2 b 2 > 0 for some suitable constant a and b, or σs) = a cosh s c,asinh s c, bs ), c with c = a 2 + b 2, and if t =2 σs) = with c = a 2 b 2 > 0or σs) = b sin s c,bcos s c, as c ), bs c,acosh s c,asinh s ), c with c = a 2 b 2 > 0 for some suitable constant a and b. Consequently, up to motions in Et 5,t=, 2, Mis an open portion of a 3- dimensional helical cylinder of the first kind or the second kind in Et 5,t=, 2, cf. Examples ). References [] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, Singapor-New Jersey-London, 984. [2] B.Y. Chen, Finite type submanifolds in pseudo-euclidean space and applications, Kodai Math. J., 8 985), [3] B.Y. Chen, Finite type pseudo-riemannian submanifolds, Tamkang J. Math., 7 986), [4] B.Y. Chen, Null 2-type surfaces E 3 are circular cylinder, Kodai Math J., 988), [5] B.Y. Chen, Null 2-type surfaces in Euclidean space, in Algebra, Analysis and Geometry, Taipei, 988) 8, World Sci. Publishing, Teaneck, NJ, 989. [6] B.Y. Chen and H. Song, Null 2-type surfaces in Minkowski space-time, Algebras, Groups and Geometries, 6 989), [7] U. Dursun, Null 2-Type Submanifolds of the Euclidean Space E 5 with Parallel Normalized Mean Curvature Vector, Kodai Math. J., ), 9 98.

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