Some Results on Conformal Geometry of Gradient Ricci Solitons
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Some Results on Conformal Geometry of Gradient Ricci Solitons J. F. Silva Filho1 Received: 28 April 2019 / Accepted: 9 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract The goal of this article is to study the conformal geometry of gradient Ricci solitons as well as the relationship between such Riemannian manifolds and closed conformal vector fields. We prove that gradient Ricci solitons endowed with a non-parallel closed conformal vector field can be conformally changed to constant scalar curvature almost everywhere. Moreover, we obtain a characterization for this class of manifolds under assumption that the closed conformal vector field is gradient type. Keywords Gradient Ricci solitons · Conformal change of metric · Conformal vector fields Mathematics Subject Classification Primary 53C20 · 53C25 · 53C44
1 Introduction In the last years there is a great interest in to study Ricci solitons, mainly motivated by the works of Hamilton and Perelman. Ricci solitons model the formation of singularities in the Ricci flow and correspond to self-similar solutions, that is, they are stationary points of this flow in the space of metrics modulo diffeomorphisms and scalings; see (Cao 2009; Chow et al. 2006; Hamilton 1995; Perelman 2002) for more details. Thus, classifying Ricci solitons or understanding their geometry is definitely an important issue. A Ricci soliton is a complete Riemannian manifold (M n , g), n ≥ 2, endowed with a smooth vector field X satisfying the fundamental equation 1 Ric + L X g = λg, 2
B 1
(1.1)
J. F. Silva Filho [email protected] Instituto de Ciências Exatas e da Natureza, Universidade da Integração Internacional da Lusofonia Afro-Brasileira, Campus das Auroras, Rua José Franco de Oliveira, 62790-970 Redenção, CE, Brazil
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where L denotes the Lie derivative and λ is a real constant. If X is the gradient vector field of a potential function f defined on M n , such a manifold is said to be gradient Ricci soliton. In particular, the fundamental Eq. (1.1) becomes Ric + H ess f = λg,
(1.2)
where H ess f stands for the Hessian of f . The Ricci soliton will be called expanding, steady or shrinking if λ < 0, λ = 0 or λ > 0, respectively. Moreover, when either the vector field X is trivial or the potential function f is constant the Ricci soliton is said to be trivial. Otherwise, it will be called of non-trivial. Notice also that Ricci solitons are natural extensions of Einstein manifolds, for more details on this subject, we recommend Besse’s book (Besse 2008) and the references therein. It is worthwhile to remark that if M n has dimension n ≥ 3 and X is a conformal vector field, then Schur’s lemma guarantees that M n is Einstein with constant scalar curvature. It is notorious the relation between conformal vector fields and Ricci solitons, especially Killing and homothetic vector fields. For instance, vector fields that provide Ricci soliton structures on a fixed Riemannian manifold are related modulo homothetic vector fields, that is, such vector fields di
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