Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 35-41, 2011 Applied Mathematics
H-Recurrent Surfaces in Euclidean SpaceEm
Kadri Arslan1, Cengizhan Murathan2, Gunay Ozturk3, Selen Turkay4 1,2Department of Mathematics Uludag University 16059 Bursa, Turkiye
e-mail: 1arslan@ uludag.edu.tr,2cengiz@ uludag.edu.tr
3Department of Mathematics Kocaeli University 41380 Kocaeli, Turkiye
e-mail: ogunay@ ko caeli.edu.tr
4Department of Mathematics Science and Technology Teachers College, Columbia
University New York , NY 10027, USA e-mail: st2282@ columbia.edu
Received Date: August 18, 2010 Accepted Date: November 13, 2011
Abstract. The object of the paper is to study some smooth surfaces M whose mean curvature vector H satisfies the H-recurrent condition DXH = λ(X)H in m-dimensional Euclidean space Em, where X is a tangent vector field of M and λ is a 1-form. First of all,we prove that the surfaces which satisfy the H-recurrent condition in Emare R⊥-parallel (i.e., R⊥H = 0). Then, we show that H-parallel surfaces in E5 are either totally umbilical or normally flat.
Key words: Mean curvature vector, H-parallel surfaces. 2000 Mathematics Subject Classification: 53C40, 53C20. 1. Introduction
Let M be a surface immersed in a Riemannian manifold Nmof dimension m. Let e
∇ denote the covariant differentiation of Nmand ξ be a normal vector field on M . If we denote by Dξ the normal component of e∇ξ, then D defines a connection in the normal bundle N (M ). A normal vector field ξ is called parallel if Dξ = 0. Let H and h denote the mean curvature vector and the second fundamental form of M in Em, respectively. It is easy to see that minimal surfaces of an Euclidean m-space Emand minimal surfaces of hyperspheres of Emare surfaces with parallel mean curvature vector of Em, i.e. DH = 0. The set of surfaces with parallel mean curvature vector in Riemannian manifold, which includes all minimal surfaces in the manifold, has been studied by many geometers. Especially, B. Y. Chen [1] and S.T. Yau [12] studied them in the case that the ambient space is an m-dimensional real space form Nm(c) of constant sectional
curvature c. They proved that if M is a surface with parallel mean curvature vector of a real space form Nm(c), then M is one of the following surfaces: (1) a minimal surface in Nm(c), (2) a minimal surface in a small hypersphere of Nm(c), and (3) a surface with constant mean curvature in a 3-sphere of Nm(c). This shows that the study of surfaces with parallel mean curvature vector in Nm(c) is reduced to that of minimal surfaces except the case (3). On the other hand, for the surfaces with parallel mean curvature vector in a complex space form see also [7].
Denote by R the curvature tensor of the van der Waerden-Bortolotti connec-tion of M and by h the second fundamental form of M . Then M is called semi-parallel if R(X, Y ) · h = 0 for all tangent vectors X and Y to M, where R(X, Y ) acts as a derivation on h. This notion is an extrinsic analogue for semi-symmetric spaces, i.e. Riemannian manifolds which satisfy R · R = 0, and a direct generalization of parallel immersions. In [3], J. Deprez showed the fact that semi-parallelity implies semi-symmetry.
For references on semi-symmetric space, see [9]. In [3], J. Deprez gave a local classification of semi-parallel hypersurfaces in Euclidean space. It is easily seen that all surfaces are semi-symmetric. In [4], J. Deprez gave a full classification of semi-parallel surfaces in Em. Deprez’s results were extended for the case the tensors R(X, Y )·h and Q(g, h) are linearly dependent, [10]. For the definition of Q, see [5].
In [8], B. Kılıç studied the surfaces M in E5satisfying the condition R(X, Y )·H = 0, where R is the curvature tensor with respect to van der Waerden-Bortolotti connection. She showed that the surfaces satisfying this condition in E5 are minimal or totally umbilical or have trivial normal connections.
In this study we consider the smooth surfaces M in m-dimensional Euclidean space Emsatisfying the H-recurrent condition
(1.1) DXH = λ(X)H,
where λ is a 1-form and H is the mean curvature vector of M . We have shown that if M is a surface immersed in Emwith H-recurrent parallel mean curvature vector, then
(1.2) R⊥(X, Y ) · H = 0.
In 2004, S. Turkay classified the surfaces immersed in E5 satisfying the condi-tion (1.2). She showed that if M is a surface satisfying condicondi-tion (1.2), then M is either minimal or totally umbilical or normally flat i.e., R⊥ = 0 [11]. Here, we give the sketch of the proof of the theorem. Recently, in [6], F. Dillen et al. generalized this result for the case M is a surface in m-dimensional real space form Nm(c) and they call these surfaces H-parallel because for these spaces the mean curvature vector H is invariant under parallel transport around in-finitesimal coordinate parallelograms. Every semi-parallel surface is H-parallel. However, the converse statement is not true in general.
2. Basic Concepts
Let M be a smooth surface immersed in Em with the Riemannian metric in-duced by the standard Riemannian metric of Em. For each p ∈ M consider the decomposition TpEm= TpM ⊕NpM , where NpM is the orthogonal complement of TpM in Em. Let e∇ be the Riemannian connection of Em. If M is a smooth surface with normal bundle N M in Em, then the connection e∇ induces the Riemannian connection ∇ on M and the connection ∇⊥ in the normal bundle N M , and for X, Y ∈ χ(M) we have
(2.1) ∇eXY = ∇XY + h(X, Y ),
where h(X, Y ) is normal to M and called the second fundamental form. Equa-tion (2.1) is known as the Gauss formula. We can further define the normal connection ∇⊥ through the following decomposition of the covariant derivative in Em of a normal vector field ξ of M in Em with respect to a tangent vector field X ∈ χ(M) into its tangential and normal components
(2.2) ∇eXξ = −AξX + ∇⊥Xξ,
where Aξ is called the shape operator with respect to ξ. The shape operator is related to the second fundamental form by g(AξX, Y ) = g(h(X, Y ), ξ), where g is the flat metric of Em. Equation (2.2) is known as the Weingarten formula. The mean curvature vector H of M is given by
(2.3) H = 1
2(h(e1, e1) + h(e2, e2)) ,
where {e1, e2, ν1, ..., νm−2} is a local orthonormal frame on M such that {e1, e2} ∈ χ(M) and {ν1, ..., νm−2} ∈ NM.
A point p of M is called umbilical if h(X, Y ) = g(X, Y )H for all X, Y ∈ TpM. Recall that a submanifold of a Riemannian manifold is said to be minimal if its mean curvature vector H vanishes identically (see, for instance, [1]). In the case under consideration, M is minimal if and only if h(e1, e1) + h(e2, e2) = 0, or equivalently < h(e1, e1) + h(e2, e2), να >= 0, f or να ∈ NM. The mean curvature vector is said to be parallel if e∇XH = 0.
Let XΛY denote the endomorphism Z →< Z, Y > X− < Z, X > Y . Then the curvature tensor R of M is given by the equation of Gauss
(2.4) R(X, Y ) =
mX−2 i=1
AiXΛAiY,
where Ai= Aνi and {ν1, ..., νm−2} is a local orthonormal basis for N(M). The
equation of Ricci becomes
(2.5) < R⊥(X, Y )ν, η >=< [Aν, Aη]X, Y > for normal vectors ν, η to M .
A surface M is said to have trivial normal connection if (2.6) R⊥(X, Y ) = DXDY − DYDX− D[X,Y ]
vanishes identically for each vector fields X and Y of χ(M ).The mean curvature vector H of a surface M is called R⊥-parallel if
(2.7) R⊥(X, Y ) · H = 0
for vectors X, Y tangent to M [6]. 3. H-recurrent Surfaces
Let M be a surface immersed in a Riemannian manifold Nmof dimension m. Let e
∇ denote the covariant differentiation of Nmand ξ be a normal vector field on M . If we denote by Dξ the normal component of e∇ξ, then D defines a connection in the normal bundle N (M ). A normal vector field ξ is called parallel if Dξ = 0. Let H and h denote the mean curvature vector and the second fundamental form of M in Em, respectively. It is easy to see that minimal surfaces of an Euclidean m-space Emand minimal surfaces of hyperspheres of Emare surfaces with parallel mean curvature vector of Em, i.e. DH = 0. B. Y. Chen [2] and S.T. Yau [12] proved the following classification theorem independently. Theorem 1. Let M be a smooth surface in m-dimensional real space form Nm(c) of constant sectional curvature c . If H is parallel in the normal bundle (i.e., DH = 0), then M is one of the following surfaces:
i) M is minimal surface in Nm(c).
ii) M is minimal surface in a small hypersphere of Nm(c).
iii) M is a surface with constant mean curvature kHk in S3of Nm(c). In the present paper, we study the smooth surfaces M satisfying the H-recurrent condition (1.1) in m-dimensional Euclidean space Em.
We prove the following result.
Theorem 2. Let M be a smooth submanifold in Em. If M satisfies the H-recurrent condition (1.1), then M is R⊥-parallel.
Proof. Assume that M satisfies H- recurrent condition (1.1). Then by using the equations (2.7) and (2.6), we get
(3.1)
R⊥(X, Y ) · H = DXDYH − DYDXH − D [X,Y ]H
= (Xλ(Y ) + λ(X)λ(Y ))H − (Y λ(X) + λ(Y )λ(X))H − λ([X, Y ])H
= (Xλ(Y ) − Y λ(X))H − λ([X, Y ])H = 2dλ(X, Y )H,
where dλ(X, Y ) is the differential of λ.
We now define a function on the subset UH = {p ∈ M | H 6= 0 at p} of M by f2= g(H, H). Using the fact that ∇g = 0, it follows from f2= g(H, H) that
(3.2) f (Y (f )) = f2λ(Y ).
So from this we have
(3.3) Y (f ) = f λ(Y ) 6= 0.
Therefore we get
(3.4) X(Y (f )) − Y (X(f)) − [X, Y ](f) = (Xλ(Y ) − Y λ(X) − λ([X, Y ]))f. Since the left hand side of above equation is identically zero and f 6= 0 by our assumption, we obtain
(3.5) dλ(X, Y ) = 0,
that is, the 1-form λ is closed. Hence from (3.5) and (3.1), we get R⊥(X, Y ).H = 0.
¤
We will use the following results.
Proposition 1. [4] Let M be a surface in Emand {ν
1, ν2, ..., νp} orthonormal vectors in N (M ) such that ν1 is in the direction of the mean curvature vector and such that Av4 = ... = Avp= 0. If we choose an orthonormal basis of T M as
eigenvectors of A1= Av1 and identify linear transformations and their matrices
in this basis, we obtain (3.6) A1= ∙ λ 0 0 μ ¸ , A2= ∙ a b b −a ¸ , A3= ∙ c d d −c ¸ ,
Proposition 2. [1] Let M be an n-dimensional submanifold of a space Nm(c) of constant sectional curvature c. Then the normal connection D is flat if and only if all of the shape operator matrices Aν are simultaneously diagonalizable. In 2004, S. Turkay ([11], pp. 55) proved the following result.
Theorem 3. Let M be a nonminimal surface in 5-dimensional Euclidean space E5. If M is R⊥-parallel, then locally
i) M is totally umbilical surface or
Proof. Let M be a smooth surface in E5such that the mean curvature vector H is parallel to ν1. If M is R⊥-parallel, then using the linearity of R⊥ we get (3.7) R⊥(ei, ej).H = αR⊥(ei, ej).ν1= 0; 1 ≤ i, j ≤ 2,
where α = √< H, H > is the mean curvature of M . Since M is not mini-mal, R⊥(e
i, ej).ν1= 0. Further, using the equations (3.6) and (2.5), after some computations, we get
b(μ − λ) = 0, d(μ − λ) = 0, from these equations the following possible cases hold:
Case I. b 6= 0, d 6= 0 and μ = λ, or Case II. b = d = 0.
For the first case M is totally umbilical. Moreover, for the second case the shape operator matrices Aνof M are simultaneously diagonalizable. So by Proposition 2, M is normally flat surface. This completes proof of the theorem.¤
Recently, in [6] the authors generalized Theorem 3 for the case M is a surface in m-dimensional real space form Nm(c). They obtained the following result: Theorem 4. [6] Let M be a H-parallel smooth surface of Nm(c). Then, M is either minimal, pseudo-umbilical or has trivial normal connection.
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