EINSTEIN SPACES AND CONFORMAL VECTOR FIELDS Dong-Soo ...

J. Korean Math. Soc. 43 (2006), No. 1, pp. 133–145

EINSTEIN SPACES AND CONFORMAL VECTOR FIELDS Dong-Soo Kim, Young Ho Kim, and Seong-Hee Park Abstract. We study Riemannian or pseudo-Riemannian manifolds which admit a closed conformal vector field. Subject to the condition that at each point p ∈ M n the set of conformal gradient vector fields spans a non-degenerate subspace of Tp M , using a warped product structure theorem we give a complete description of the space of conformal vector fields on arbitrary non-Ricci flat Einstein spaces.

1. Introduction In 1925, Brinkmann studied conformal mappings between Riemannian or pseudo-Riemannian Einstein spaces ([1]). Later conformal vector fields, or infinitesimal conformal mappings on Einstein spaces were reduced to the case of gradient vector fields, leading to a very fruitful theory of conformal gradient vector fields in general. Brinkmann’s work has attracted renewed interest, especially in the context of general relativity ([2, 3, 4, 5, 9, 14, 17]), and the following local structure theorem has been shown: Proposition 1.1. (Kerckhove [9]) Let (M n , g) be an n-dimensional Einstein but not Ricci flat pseudo-Riemannian manifold with Ric = (n − 1)kg(k 6= 0), which carries a conformal vector field. Here we denote by Ric the Ricci tensor of (M n , g). Suppose that the subspace ∆(p) spanned by the set of conformal gradient vector fields at p ∈ M n satisfies the following: (a) ∆(p) is a non-degenerate subspace of Tp M for each p ∈ M, Received February 6, 2005. Revised July 19, 2005. 2000 Mathematics Subject Classification: 53C50, 53A30, 53C25. Key words and phrases: Einstein space, warped product, conformal vector field. This study was financially supported by Chonnam National University in the program, 2002.

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(b) the dimension m of ∆(p) is independent of the choice of the point p. Then (M n , g) is locally isometric to a warped product B m (k)×f F, where B m (k) is an m-dimensional space of constant sectional curvature k and the fibre (F, gF ) is an Einstein space with RicF = (n − m − 1)αgF for some constant α. Generalizing Kerckhove’s results, the first two authors established a local structure theorem as follows: Proposition 1.2. ([11]) Let (M n , g) be an n-dimensional connected pseudo-Riemannian manifold. Suppose that there exists a nonzero constant k ∈ R such that (a) dim Ak (M n , g) = m ≥ 1, (b) each subspace ∆(p) is a nondegenerate subspace of Tp M . Then, for a fixed po ∈ M n the following hold: (1) if dim ∆(po ) < m, then (M n , g) is locally isometric to a space form B n (k), (2) if dim ∆(po ) = m, then (M n , g) is locally isometric to a warped product space B m (k) ×σT F, where σT is the height function for a vector T in the pseudo-Euclidean space of B m (k) and the fiber (F n−m , gF ) is a pseudo-Riemannian manifold. Furthermore, F satisfies the following: (i) if hT, T i 6= 0, then Aα (F, gF ) = {0}, where α = hT, T i, (ii) if hT, T i = 0, then F carries no nontrivial homothetic gradient vector fields. In either case, we have Ak (M n , g) = {˜ σS |σS ∈ Ak (B m (k)), hS, T i = 0}, where σ ˜S denotes the lifting of σS . For the definitions of Ak (M n , g), ∆(p) and σT , see Section 2. Note that if (F, gF ) has constant sectional curvature α = hT, T i, then the warped product M n = B m (k) ×σT F has constant sectional curvature k. This shows that the space of closed conformal vector fields on M n = B m (k) ×σT F is of dimension n + 1, which is greater than the dimension m of {˜ σS |σS ∈ Ak (B m (k)), hS, T i = 0}. This shows that not all closed conformal vector fields on the warped product need to be lifted from the base. On the other hand, the first two authors prove as follows that the necessary condition on the fibre F in (2) of Proposition 1.2 is sufficient for any closed conformal vector fields on the warped product M n = B m (k) ×σT F to be lifted from the base.

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Proposition 1.3. ([11]) Let (M n , g) be a warped product space B (k) ×σT F , where T is a vector in the ambient pseudo-Euclidean space of B m (k). Suppose that the fibre (F, gF ) satisfies the following: (1) if hT, T i 6= 0, then Aα (F, gF ) = {0}, where α = hT, T i, (2) if hT, T i = 0, then F carries no nontrivial homothetic gradient vector fields. Then (M n , g) satisfies the following: m

Ak (M n , g) = {˜ σS |σS ∈ Ak (B m (k)), hS, T i = 0}. In particular, each subspace ∆(p) is nondegenerate and of dimension m. In this paper, we study Riemannian or pseudo-Riemannian manifolds which admit a closed conformal vector field. Subject to the condition that at each point p ∈ M n the set of conformal gradient vector fields spans a non-degenerate subspace of Tp M , using Proposition 1.2 and Proposition 1.3 we give a complete description of the space of conformal vector fields on arbitrary non-Ricci flat Einstein spaces (Theorem 3.3). 2. Closed conformal vector fields We consider an n-dimensional connected pseudo-Riemannian manifold (M n , g) carrying a closed conformal vector fields V. Hence there is a smooth function φ on M n such that (2.1)

∇X V = φX

for all X ∈ T M, where ∇ denotes the Levi-Civita connection on M n . Then for every point p ∈ M n one can find a neighborhood U and a function f such that V = ∇f, the gradient of f. Hence the Hessian ∇2 f and the Laplacian ∆f of f are given respectively by (2.2)

∇2 f = φg,

∆f = divV = nφ.

n

We denote by CC(M , g) the vector space of closed conformal vector fields. First of all, we state some useful lemmas for later use. Lemma 2.1. Let V be a non-trivial closed conformal vector field. (1) If γ : [0, `) → M n is a geodesic with V (γ(0)) = aγ 0 (0) for some a ∈ R, then we have Z t φ(γ(s))ds)γ 0 (t). (2.3) V (γ(t)) = (a + 0

(2) If V (p) = 0, then divV (p) = nφ(p) 6= 0, in particular, all zeros of V are isolated.

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Proof. See Propositions 2.1 and 2.3 in [16].

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Lemma 2.2. Let (M n , g) be an n-dimensional connected pseudoRiemannian manifold. Then the following hold: (1) dim CC(M n , g) ≤ n + 1, (2) if dim CC(M n , g) ≥ 2, there exists a constant k ∈ R such that for all V ∈ CC(M n , g), ∇φ = −kV, where nφ is the divergence of V. Proof. See Proposition 2.3 in [16] and Proposition 4 in [7].

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The model spaces B n (²a2 )(² = ±1) are the hyperquadrics Sνn (a2 ), which are given by

Hνn (−a2 ),

Sνn (a2 ) = {x ∈ Rn+1 |hx, xi = 1/a2 }, ν 2 Hνn (−a2 ) = {x ∈ Rn+1 ν+1 |hx, xi = −1/a }.

For a fixed vector T in Rn+1 or Rn+1 ν ν+1 , let σT be the height function in the direction of T defined by σT (x) = hT, xi. Then one can easily show that on B n (k), (2.4)

∇σT (x) = T − kσT (x)x,

(2.5)

∇X ∇σT = −kσT X

for all X ∈ T B n (k)([9]). (2.5) implies that for any constant vector T in Rn+1 or Rn+1 ν ν+1 , ∇σT is a closed conformal vector field on the hypern quadric B (k), k = ²a2 . Furthermore, by counting dimensions, we see that ∇σT represents every element of CC(B n (k)). For the flat space form Rnν with index ν, the vector field V defined by V (x) = bx + c(b ∈ R, c ∈ Rnν ) is a closed conformal vector field. Obviously, by counting dimensions, we have CC(Rnν ) = {bx + c|b ∈ R, c ∈ Rnν }. For the space of conformal vector fields of pseudo-Riemannian space forms, the authors et al. gave a complete description in [12]. Now we introduce a function space Ak (M n , g)(k 6= 0) and a symmetric bilinear form Φk on the space as follows: (2.6)

Ak (M n , g) = {f ∈ C ∞ (M )|∇X ∇f = −kf X,

X ∈ T M },

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(2.7)

Φk (f, h) = h∇f, ∇hi + kf h,

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f, h ∈ Ak (M n , g).

For the non-flat space form B n (k)(k = ²a2 ) (2.4) shows that (2.8)

Φk (σT , σS ) = h∇σT , ∇σS i + kσT σS = hT, Si.

This implies that the symmetric bilinear form Φk is just the usual scalar product on the ambient pseudo-Euclidean space. Recall that LV g denotes the Lie derivative of g with respect to V. Lemma 2.3. ([21]) Let (M n , g) be a totally umbilic submanifold of a ¯ , g¯). If V is a conformal vector field on M ¯ pseudo-Riemannian space (M T n with LV g = 2σg, then the tangential part V of V on M is a conformal vector field on M n with LV T g = 2{σ + g¯(V, H)}g, ¯. where H denotes the mean curvature vector field of M n in M In [12], the authors et al. prove a converse of Lemma 2.3 for hypersurfaces of a pseudo-Riemannian space form. For a point p ∈ M n , let ∆(p) denote the span of the set of closed conformal vector fields at p, that is, ∆(p) = {V (p) ∈ Tp (M )|V ∈ CC(M n , g)}. Suppose that CC(M n , g) is of dimension m ≥ 2. Then (2.1) and Lemma 2.2 imply that there exits a constant k ∈ R such that for all V ∈ CC(M n , g) with φ = (1/n)divV (2.9)

∇φ = −kV,

so that we have (2.10)

∇X ∇φ = −kφX, X ∈ T M.

Hence, if k is nonzero, then CC(M n , g) and ∆(p) may be identified with Ak (M n , g) and {∇f (p)|f ∈ Ak (M n , g)}, respectively.

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3. Einstein spaces admitting conformal vector fields Let (M n , g) denote a non-Ricci flat connected Einstein space with Ric= k(n − 1)g(k 6= 0) which admits conformal vector fields. Recall that by definition the Lie derivative of the metric g is given by LV g(X, Y ) = g(∇X V, Y ) + g(X, ∇Y V ) for arbitrary tangent vectors X, Y , where ∇ denotes the Levi-Civita connection. We denote by C(M n , g) the Lie algebra of all conformal vector fields on (M n , g), and by I(M n , g) the subalgebra of isometric vector fields. It is well known ([8, 22]) that if V is a conformal vector field on M n with LV g = 2f g, then (3.1)

∇X ∇f = −kf X

for all X ∈ T M. Hence we see that kV + ∇f is an isometric vector field. Since k 6= 0, this together with (3.1) shows that C(M n , g) may be identified with I(M n , g) ⊕ Ak (M n , g) by the correspondence V → (kV + ∇f, f ). If V is a closed conformal vector field on M n with ∇X V = f X, X ∈ T M , then (3.1) shows that w = kV + ∇f is parallel on M n . Since (M n , g) is Einstein with nonzero scalar curvature k, w must be trivial, that is, V = − k1 ∇f. Hence CC(M n , g) can be identified with Ak (M n , g) by the correspondence V → f . Thus ∆(p) is the subspace of Tp M spanned by ∇f (p), f ∈ Ak (M n , g). Using Proposition 1.2, we can improve the Kerkhove’s results (Proposition 1.1) as follows: Proposition 3.1. Let (M n , g) be a non-Ricci flat connected Einstein space with Ric = (n − 1)kg(k 6= 0) which admits a nonisometric conformal vector field. We denote by m the dimension of the space Ak (M n , g). Suppose that each subspace ∆(p) is a nondegenerate subspace of Tp M . Then for a fixed point po ∈ M n one of the following holds: (1) if dim ∆(po ) < m, then (M n , g) is locally isometric to B n (k). (2) if dim ∆(po ) = m, then (M n , g) is locally isometric to a warped product space B m (k)×σT F, where σT ∈ Ak (B m (k)) and the fibre (F, gF ) is an (n − m)-dimensional Einstein space with RicF = (n − m − 1)hT, T igF . Furthermore, F satisfies the following: (i) if hT, T i 6= 0, then F admits no nonisometric conformal vector fields,

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(ii) if hT, T i = 0, then F admits no nontrivial homothetic gradient vector fields. In either case, Ak (M n , g) = {˜ σS |σS ∈ Ak (B m (k)), hS, T i = 0}, where σ ˜S denotes the lifting of σS . Proof. Since (M n , g) admits a nonisometric conformal vector field, the space Ak (M n , g) is of dimension m ≥ 1. Hence the proof follows from Proposition 1.2 except the condition on the fibre F with hT, T i = 6 0. Furthermore, the fibre (F, gF ) is an (n − m)-dimensional Einstein space with RicF = (n − m − 1)αgF , where α denotes hT, T i. Since α 6= 0, (3.1) shows that Aα (F, gF ) = {0} is equivalent to the condition that F carries no nonisometric conformal vector fields. ¤ Remark. Suppose that n−m−1 ≤ 0 holds in Proposition 3.1, that is, the dimension m of CC(M n , g) is greater than or equal to n − 1. Then the proof of Proposition 2.3 in [11] shows that (M n , g) has constant sectional curvature k. Thus (M n , g) is locally isometric to B n (k). In particular, we have m = n + 1. This shows that if (M n , g) is not locally isometric to B n (k), then we have m < n − 1, that is, n − m − 1 ≥ 1. Suppose that the fibre F in Proposition 3.1 satisfies hT, T i = 0. Then F carries no nontrivial homothetic gradient vector fields. For any conformal vector field V on F with LV gF = 2f gF , (3.1) shows that ∇f is isometric, hence it is a homothetic gradient vector field. Thus ∇f vanishes identically, that is, V is a homothetic vector field. Therefore the fiber space F carries no nonhomothetic conformal vector fields. The following lemma is useful for the proof of our theorem. Lemma 3.2. Suppose that (M n , g) is a non-Ricci flat Einstein space with Ric = k(n − 1)g(k 6= 0). Then, for any w ∈ I(M n , g) and σ ∈ Ak (M n , g), we have hw, ∇σi ∈ Ak (M n , g) and ∇hw, ∇σi = [w, ∇σ]. Proof. For the proof, see p.18 in [8].

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Now let’s describe the space C(M n , g) of a non-Ricci flat Einstein warped product space (M n , g) = B m (k)×σT F (k 6= 0) in Proposition 3.1. Since the space C(M n , g) can be identified with I(M n , g) ⊕ Ak (M n , g) and the function space Ak (M n , g) is given by Proposition 1.3, it suffices to describe the space I(M n , g) of isometric vector fields. Theorem 3.3. Let (M n , g) be a non-Ricci flat Einstein warped product space B m (k) ×σT F (k 6= 0), where (F, gF ) is an (n − m)-dimensional

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Einstein space with RicF = (n − m − 1)hT, T igF . Suppose that the fiber F satisfies the following: (i) if hT, T i 6= 0, then F admits no nonisometric conformal vector fields, (ii) if hT, T i = 0, then F admits no nontrivial homothetic gradient vector fields. Then the following hold: (1) if hT, T i = 0 and F admits a nonisometric homothetic vector field, then M n admits an isometric vector field wo with hwo , ∇σT i = σT and we have I(M n , g) = Rwo ⊕ {w ˜1 |w1 ∈ I(B m (k)), hw1 , ∇σT i = 0} ⊕ {w ˜2 |w2 ∈ I(F, gF )}, (2) otherwise, we have I(M n , g) ={w ˜1 |w1 ∈ I(B m (k)), hw1 , ∇σT i = 0} ⊕ {w ˜2 |w2 ∈ I(F, gF )}, where for each i = 1, 2, w ˜i denotes the lifting of wi . Proof. First, note that the condition (ii) on F implies that F admits no nonhomothetic conformal vector fields. Furthermore, it follows from Proposition 1.3 that (M n , g) satisfies the following: (3.2)

Ak (M n , g) = {˜ σS |σS ∈ Ak (B m (k)), hS, T i = 0},

where σ ˜S denotes the lifting of σS . For σ ˜S ∈ Ak (M n , g) with S 6= 0, (2.4) and Φk (σS , σT ) = hS, T i = 0 imply that ∇σS (p) 6= 0 for any p ∈ B m (k). This means that for each p ∈ B m (k), {∇σS (p)|hS, T i = 0} spans Tp B m (k). Now we give a proof of Theorem 3.3 step by step in the following procedures (a)–(d). (a) For every w ∈ I(M n , g) we have hw, ∇σT i = cσT for some constant c in case (1) and we have hw, ∇σT i = 0 in case (2). For each p ∈ B m (k), consider the tangential part wF of w on the fiber {p} × F. Since {p} × F is a totally umbilic submanifold of M n with mean curvature vector field − σ1T ∇σT , we obtain from Lemma 2.3 (3.3)

LwF gp = −

2 hw, ∇σT igp , σT

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where gp denotes the restriction of g to the fibre {p}×F. Because {p}×F is homothetic to F and F admits no nonhomothetic conformal vector fields, (3.3) shows that hw, ∇σT i = σT h for some function h on B m (k). Hence in case (2), it follows again from (3.3) that h must be trivial. In case (1), using Lemma 3.2, we can prove that h is constant. In fact, since hT, T i = 0, we have hw, ∇σT i = σT¯ for a vector T¯ with hT, T¯i = 0. Let S be an arbitrary vector with hS, T i = 0. Then we ¯ T i = 0. Because w is also have hw, ∇σS i = σS¯ for a vector S¯ with hS, n isometric and σS , σS¯ , σT belongs to Ak (M , g), we obtain ∇σS (σT¯ ) = ∇σS hw, ∇σT i = h∇∇σS w, ∇σT i + hw, ∇∇σS ∇σT i (3.4)

= −h∇σS , ∇∇σT wi − kσT σS¯ = −∇σT h∇σS , wi + h∇∇σT ∇σS , wi − kσT σS¯ = −∇σT (σS¯ ) − kσS hw, ∇σT i − kσT σS¯ = −kσS σT¯ ,

¯ T i = 0. It follows from (3.4) where the last equality follows from hS, ¯ that hS, T i = 0. This together with hS, T i = 0 implies ∇σS (h) = ∇σS (

σT¯ ) = 0. σT

This completes the proof of (a) because ∇σS (p) with hS, T i = 0 generates the tangent space of B m (k) at each point p ∈ B m (k). (b) For a linear map Ψ from I(B m (k)) into Ak (B m (k)) defined by Ψ(w) = hw, ∇σT i, the image of Ψ is the subspace {σS |hS, T i = 0}. First of all, we show that the image of Ψ is contained in the subspace {σS |hS, T i = 0}. It is easily seen that Ψ is well-defined by Lemma 3.2. Since hw, ∇σT i belongs to Ak (B m (k)), we see that hw, ∇σT i = σS for some S in the ambient pseudo-Euclidean space. Furthermore, we have ∇σS = [w, ∇σT ] by Lemma 3.2. Since σT ∈ Ak (B m (k)) we have ∇w ∇σT = −kσT w. On the other hand, ∇w ∇σT = ∇∇σT w +[w, ∇σT ] = ∇∇σT w + ∇σS , so that we obtain (3.5)

∇∇σT w + ∇σS + kσT w = 0.

Since w is isometric, taking the scalar products of the both sides of (3.5) with ∇σT , we see that Φk (σS , σT ) = h∇σS , ∇σT i + kσS σT = 0.

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This shows that the image of Ψ is contained in the m-dimensional subspace {σS |hS, T i = 0}. Let w ∈ KerΨ, that is, hw, ∇σT i = 0. For a fixed point p ∈ B m (k), (3.5) implies that (3.6)

hw(p), ∇σT (p)i = 0, ∇∇σT (p) w = −kσT (p)w(p).

Conversely, if w satisfies (3.6), then (3.5) together with (3.6) shows that the function σS = hw, ∇σT i satisfies σS (p) = 0 and ∇σS (p) = 0. Since σS ∈ Ak (B m (k)), Lemma 2.1 yields that σS = hw, ∇σT i vanishes identically. This means that w ∈ KerΨ is characterized by (3.6). Hence the proof of Lemma 28 ([18], p.253) gives dimKerΨ ≤ m(m − 1)/2. Since dimI(B m (k)) = m(m + 1)/2, we see that ImΨ is of at least mdimensional. This completes the proof of (b). (c) If hT, T i = 0 and F carries a nonisometric homothetic vector field u, then there exists an isometric vector field wo ∈ I(M n , g) which satisfies hwo , ∇σT i = σT . Furthermore, we have I(M n , g) = Rwo ⊕ {w ∈ I(M n , g)|hw, ∇σT i = 0}. Since hT, T i = 0, (b) shows that there exists an isometric vector field w1 on B m (k) such that hw1 , ∇σT i = σT on B m (k). We may assume that the nonisometric homothetic vector field u satisfies Lu gF = 2gF . We put wo = w ˜1 − u ˜, where w ˜1 and u ˜ denote the liftings of w1 and u, respectively. Then it is easy to prove that wo is an isometric vector field on M n which satisfies hwo , ∇σT i = σT on M n . For any w ∈ I(M n , g), hw, ∇σT i = cσT for some constant c. It gives hw − cwo , ∇σT i = 0. This completes the proof of (c). (d) For w ∈ I(M n , g), we consider the decomposition w = wB + wF , where wB and wF are tangent to the leaves and to the fibers of M n , respectively. Suppose that w satisfies hw, ∇σT i = 0. Then wB and wF are isometric vector fields on M n . Furthermore, wB and wF are liftings of w1 ∈ I(B m (k)) and w2 ∈ I(F, gF ), respectively. Obviously, w1 satisfies hw1 , ∇σT i = 0. ˜ Y˜ and U ˜ , V˜ of X, Y ∈ T B m (k) and U, V ∈ T F, For any liftings X, ˜ Y˜ ) = 0. hw, ∇σT i = 0 gives respectively, one can see that LwF g(X, ˜ ˜ ˜ Y˜ ) = 0 and LwF g(U ˜ , V˜ ) = 0 LwB g(U , V ) = 0. Hence we have LwB g(X, ˜ ˜ ˜ ˜ wi to because w is isometric. We need to prove LwB g(X, U ) = U hX,

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˜ U ˜ ) = 0 holds. Since ∇σS (σS ∈ Ak (M n , g)) genshow that LwB g(X, erates the tangent space of the base B m (k), it suffices to show that ˜ h∇σS , wi = 0 for all σS ∈ Ak (M n , g). Lemma 3.2 implies that h∇σS , U wi belongs to Ak (M n , g), and hence (3.2) shows that it is a function on the base B m (k). This shows that wB (and hence wF ) is isometric on M n. Now we fix a point qo ∈ F. Let w1 be the restriction wB |B×{qo } of B w . Since hw1 , ∇σT i = 0, we see that the lifting w ˜1 is an isometric vector field on M n . Let η1 be the isometric vector field wB − w ˜1 on M n . From the following: ˜ U ˜) = U ˜ hX, ˜ wB i = 0, LwB g(X, ˜ U ˜) = U ˜ hX, ˜ w Lw˜1 g(X, ˜1 i = 0, ˜ hX, ˜ η1 i = 0 for any X ∈ T B m (k) and U ∈ T F. Since we see that U ˜ η1 i = 0 on B × {qo }, hX, ˜ η1 i is identically zero on M n . Hence η1 = hX, B w −w ˜1 vanishes identically on M n . For a fixed point po ∈ B m (k), let w2 be the restriction wF |{po }×F of wF . Then the lifting w ˜2 is isometric on M n . Let η2 denote the isometric F vector field w − w ˜2 . For any σS ∈ Ak (M n , g) it is easy to show the following: ˜ ) = ∇σS hwF , U ˜ i − 2 h∇σS , ∇σT i hwF , U ˜ i = 0, LwF g(∇σS , U σT ˜ ) = ∇σS hw ˜ i − 2 h∇σS , ∇σT i hw ˜ i = 0. Lw˜2 g(∇σS , U ˜2 , U ˜2 , U σT Hence, from Φk (σS , σT ) = 0 we obtain for all σS ∈ Ak (M n , g) ˜ i + 2kσS hη2 , U ˜ i = 0. (3.7) ∇σS hη2 , U Since ∇σS , σS ∈ Ak (M n , g) generates the tangent spaces of leaves and ˜ i = 0 on {po } × F, (3.7) implies that hη2 , U ˜ i = 0 for any U ∈ T F, hη2 , U F and hence η2 = w − w ˜2 vanishes identically on M n . In fact, let γ(t) be an integral curve of ∇σS on M n with an initial point on {po } × F and h ˜ i. Then (3.7) shows that the function y(t) = denote the function hη2 , U h(γ(t)) satisfies the first order differential equation y 0 (t)+2kf (t)y(t) = 0, where f (t) denotes σS (γ(t)). Because of the initial condition y(0) = 0, the function y(t) vanishes identically. This completes the proof of (d). Combining from (a) to (d), the proof of our theorem is completed. ¤ Acknowledgements. The authors would like to express their deep thanks to the referee for valuable suggestions to improve the note.

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Dong-Soo Kim Department of Mathematics Chonnam National University Kwangju 500-757, Korea E-mail : [email protected] Young Ho Kim Department of Mathematics Kyungpook National University Daegu 702-701, Korea E-mail : [email protected] Seong-Hee Park Department of Mathematics Chonnam National University Kwangju 500-757, Korea E-mail : [email protected]