PDF《Turk J Math》

Turk J Math (2017) 41:

1365 C

1375 c ? T¨ UBB ITAK doi:10.

3906/mat-1606-49 Warped product spaces with Ricci conditions Sang Deok LEE1 , Byung Hak KIM2,? , Jin Hyuk CHOI3

1 Department of Mathematics, Dankook University, Cheonan, Korea

2 Department of Applied Mathematics, Kyung Hee University, Yongin, Korea

3 Humanitas College, Kyung Hee University, Yongin, Korea Received: 10.06.2016 ? Accepted/Published Online: 03.01.2017 ? Final Version: 23.11.2017 Abstract: In this paper, we study the Ricci soliton in the Riemannian products M = Rn * B and warped products M = R *f B of the Euclidean space and Riemannian manifolds, and the gradient Ricci soliton in the warped products M = S1 *f B of 1-dimensional circle and Riemannian manifolds. Moreover, we introduce the concept of the generalized Ricci soliton and we suggest the method of construction of the Riemannian manifold (M, g) with a Ricci soliton g. Finally, we also study the Lorentzian warped products with the Ricci soliton. Key words: Ricci curvature, warped product space, Ricci soliton 1. Introduction The concept of a Ricci soliton was introduced by Hamiliton [7], which is both a generalization of the Einstein metric and a special solution of the Ricci ?ow. A Riemannian metric g on a complete Riemannian manifold M is called a Ricci soliton if there exists a smooth vector ?eld X such that the Ricci tensor satis?es the following equation: Ric +

1 2 LXg = ρg (1.1) for some constant ρ, where LX is the Lie derivative with respect to X [2, 3, 5, 6, 9]. It is said that (M, g) or M is a Ricci soliton if the metric g on M is a Ricci soliton. The Ricci soliton is called shrinking if ρ >

0, steady if ρ = 0, and expanding if ρ <

0. The metric of a Ricci soliton is useful in not only physics but also mathematics, and it is often referred to as quasi-Einstein [4]. If X = ?h for some function h on M , then M is called a gradient Ricci soliton. In this case, equation (1.1) can be rewritten as: Ric + Hess h = ρg, (1.2) and h is called a potential function. It is well known that when ρ ≤

0 all compact solitons are necessarily Einstein [6], and a Ricci soliton on a compact manifold has a constant curvature in

2 dimensions [7] as well as in

3 dimensions [8]. Moreover, a Ricci soliton on a compact manifold is a gradient Ricci soliton [9, 14] and Vaghef and Razavi [15] studied the stability of compact Ricci solitons under Ricci ?ow. ?Correspondence: [email protected]

2010 AMS Mathematics Subject Classi?cation: Primary 53C25;

Secondary 53B21

1365 LEE et al./Turk J Math In [12], we studied the gradient Ricci soliton in the warped product space and obtained the criterion for the base space of the warped product space with a gradient Ricci soliton to be a gradient Ricci soliton or an Einstein space by taking the derivative twice of the warping function. However, there are no such criteria or methods of construction of a Ricci soliton for the Riemannian products or warped products as far as we know. Thus, it is natural to consider the case of noncompact Riemannian products or warped product spaces with a Ricci soliton. From this point of view, we study the Riemannian product space M = Rn *Bm with a Ricci soliton and obtain the fact that M is a Ricci soliton if and only if B is a Ricci soliton. Furthermore, we get a necessary condition for the base space to be a Ricci soliton in the warped products R*f B with a Ricci soliton. Moreover, we introduce a generalized Ricci soliton with the Ricci soliton warping function f , and using Theorem 3.7 we can suggest the method of construction of the warped products R *f B to admit a Ricci soliton. Precisely, we proved that if B is a generalized Ricci soliton with the Ricci soliton warping function f , then M = R *f Bm is a Ricci soliton. On the other hand, Kenmotsu [10] gave a characterization of the warped product space L *f CEn by tensor equations. By use of the almost contact structure on L *f CEn introduced by Kenmotsu [10] and combining our theorems, we see that the warped product space M = R*f B with Ricci soliton for the structure vector is, in fact, Einstein and also B is Einstein, where B is a Kaehler manifold and f = cet for a constant c. In the consideration of the warped product space M = S1 (k) *f B with a gradient Ricci soliton, we studied the relationship of the warping function f and the Einstein metric on B. Moreover, we clarify the function h appearing in equation (1.2) for the warped products M = S1 *f B. Finally, we study the Lorentzian warped product space R *f B with a Ricci soliton and obtain the necessary condition of the base space to be a Ricci soliton. For the converse of this case, we can construct the Lorentzian warped product space R *f B admitting a Ricci soliton when the base space is a generalized Lorentzian Ricci soliton with a Lorentzian Ricci soliton warping function f . Consequently, it is possible to construct a Ricci soliton on the Riemannian product space or the warped product space by use of our results, and not only the Riemannian case but also the Lorentzian case. 2. Ricci solitons in Riemannian product manifolds Let (B, g) be an m-dimensional Riemannian manifold with a metric g and let M = Rn *B be the Riemannian product manifold with the metric ? g = ( δuv