PDF《Turk J Math》

0 0 gab ) , where the range of indices u, v, w, ・ ・ ・ is {1, 2,n} and the range of indices a, b, c, ・ ・ ・ is {n + 1,n + m}. Then the Ricci curvature tensors ? S and S of M and B, respectively, are given by ? Sab = Sab and the others are zero. Suppose that B is a Ricci soliton. If we take ρ = ρ and a covector ?eld ? U = (? ξu, ? ξa) on M by ? ξu = ρtu, ? ξa = ξa on B, then we obtain ? Sab = Sab = ρgab ?

1 2 (?aξb + ?bξa) = ? ρ? gab ?

1 2 ( ? ?a ? ξb + ? ?b ? ξa), ? Sau =

0 = ?1

2 ( ? ?a ? ξu + ? ?u ? ξa), ? Suv =

0 = ? ρδuv ?

1 2 ( ? ?u ? ξv + ? ?v ? ξu). (2.1) Hence, we can see that if B is a Ricci soliton, then M = Rn * B is a Ricci soliton.

1366 LEE et al./Turk J Math Conversely, if M = Rn * B is a Ricci soliton, then there exists a covector ?eld ? V on M such that Sij = ρ gij ?

1 2 (?i ? ξj + ?j ? ξi) for some constant function ? ρ on M , where ? ξi is a dual component of smooth vector ?eld ? V on M and the range of indices i, j, k, ・ ・ ・ is {1, 2, 3,n, n + 1,n + m}. Thus, we have Sab = ? Sab = ? ρgab ?

1 2 (?a ? ξb + ?b ? ξa ?

2 { c ab } ξc), ?a ? ξu + ?u ? ξa = 0, ? ρδuv ?

1 2 (?u ? ξv + ?v ? ξu) = 0. (2.2) From the third equation of (2.2) with the case u = v, we obtain ? ξu = ? ρxu + hu (u), (2.3) where hu (u) is a function on M having no xu -variable. The third equation of (2.2) with the case u ?= v and (2.3) give rise to ?uhv (v) = ??vhu (u), (2.4) which means that ?uhv (v) is a function on M having no xu -variable and xv -variable. Thus, we can put H(u,v) = ?uhv (v). Integrating this equation and using (2.3), we get hu (u) = ?H(u,v)xv + ku (v), (2.5) where ku (v) is a function on M having no xv -variable. Then from equations (2.3) and (2.5), we obtain ? ξu = ? ρxu ? ∑ u?=v H(u,v)xv + Ku (1,2,・・・ ,n), (u ?= v). (2.6) On the other hand, the second equation of (2.2) and (2.6) give ? ξa = ? n ∑ u=1 ?aKu (1,2,・・・ ,n)xu + La (1,2,・・・ ,n) (2.7) and the ?rst equation of (2.2) gives Sab ? ? ρgab = ?

1 2 (?a ? ξb + ?b ? ξa ?

2 { c ab } ? ξc). Hence, if we consider equations (2.6) and (2.7), we see that ? V = (? ρx1 ? ∑ 1?=v H(1,v)xv + K1 (1,2,・・・ ,n) ρxn ? ∑ n?=v H(n,v)xv + Kn (1,2,・・・ ,n),

1367 LEE et al./Turk J Math ? n ∑ u=1 ?n+1Ku (1,2,・・・ ,n)xu + Ln+1 (1,2,・・・ ,n) n ∑ u=1 ?n+mKu (1,2,・・・ ,n)xu + Ln+m (1,2,・・・ ,n)). If we take V = (ξn+1,ξn+m) such that ξa = La (1,2,・・・ ,n) , then we get Sab ? ρgab = ?

1 2 (?aLb (1,2,・・・ ,n) + ?bLa (1,2,・・・ ,n)) for ρ = ? ρ, because V is a smooth vector ?eld on B and Sab ? ρgab is independent of Rn . Hence, we can state that if M = Rn * B is a Ricci soliton, then B is a Ricci soliton. Thus, we have: Theorem 2.1 M = Rn * B is a Ricci soliton if and only if B is a Ricci soliton. 3. Ricci solitons in warped product manifolds Consider the warped product space M = R *f B with ? g = (

1 0

0 f2 g ) , where f : R → R+ is a warping function and g is a Riemannian metric on B. Then the Ricci curvature te........