m -Quasi-Einstein Metrics Satisfying Certain Conditions on the Potential Vector Field
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m-Quasi-Einstein Metrics Satisfying Certain Conditions on the Potential Vector Field Amalendu Ghosh Abstract. In this paper we study Riemannian manifolds (M n , g) admitting an m-quasi-Einstein metric with V as its potential vector field. We derive an integral formula for compact m-quasi-Einstein manifolds and prove that the vector field V vanishes under certain integral inequality. Next, we prove that if the metrically equivalent 1-form V associated with the potential vector field is a harmonic 1-form, then V is an infinitesimal harmonic transformation. Moreover, if M is compact then it is Einstein. Some more results were obtained when (i) V generates an infinitesimal harmonic transformation, (ii) V is a conformal vector field. Mathematics Subject Classification. 53C25, 53C20, 53D10. Keywords. Ricci soliton, m-quasi-Einstein metric , infinitesimal harmonic transformation , conformal vector field.
1. Introduction In the recent years, Einstein metrics and several of their generalizations [1] have received a lot of importance in geometry and physics. These are Ricci solitons, Ricci almost solitons, m-quasi-Einstein metrics and generalized quasi-Einstein metrics. Ricci solitons have been extensively studied, also because of their connection with the study of the Ricci flow. A Ricci soliton is a Riemannian manifold (M n , g) together with a vector field V that satisfies £V g + 2S = 2λg, where £V denotes the Lie-derivative operator along a vector field V , and S the Ricci tensor of g and λ a constant. It is said to be trivial (Einstein) if either V = 0, or V is Killing. This is said to be a gradient Ricci soliton if V = Df , for some smooth function f on M , where D is the gradient operator. For details about Ricci soliton, we refer to [2]. Generalizing the notion of gradient Ricci soliton, Case et al. [3] introduced the notion quasi-Einstein metric. This is closely related to the warped 0123456789().: V,-vol
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product spaces (see [1]) and appears from the m-Bakry-Emery Ricci tensor Sfm , defined by (see [4]) 1 df ⊗ df. m A Riemannian manifold M together with a Riemannian metric g is said to quasi-Einstein respect to the function f and the constant m if Sfm = λg, i.e., if its Ricci tensor S satisfies 1 (1.1) S + ∇2 f − df ⊗ df = λg, m where λ is a constant, 0 < m ≤ ∞ and ∇2 f denotes the Hessian tensor of the smooth function f on M . This also appears from the warped product of the base of an (n + m)-dimensional Einstein manifold (see [5]). If λ is a smooth function in the defining condition (1.1), then it is known as generalized mquasi-Einstein. For more details we refer to [6–8]). Equation (1.1) reduces to the usual Einstein condition when f is constant. Moreover, when m = ∞ it reduces to exactly the gradient Ricci soliton [2]. Thus, Eq. (1.1) can be regarded as a generalization of gradient Ricci soliton. In [3], several results were proved extending rigidity results for gradient Ricci solitons presented by Petersen–Wylie [9]. Recently, Nurowski and Randall [10] extended the notion of Ricci soli
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