On gradient Schouten solitons conformal to a pseudo-Euclidean space
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Romildo Pina
© Springer-Verlag GmbH Germany, part of Springer Nature 2019
· Ilton Menezes
On gradient Schouten solitons conformal to a pseudo-Euclidean space Received: 14 March 2019 / Accepted: 10 October 2019 Abstract. In this paper we consider ρ-Einstein solitons that are conformal to a pseudoEuclidean space and invariant under the action of the pseudo-orthogonal group. We provide all the solutions for the gradient Schouten soliton case. Moreover, we proved that if a gradient Schouten soliton is both complete, conformal to a Euclidean metric, and rotationally symmetric, then it is isometric to R × Sn−1 .
1. Introduction and main statements In 1982 Hamilton [1] introduced the Ricci flow equation ∂ g(t) = −2Ric(g(t)) ∂t where Ric(g(t)) is the Ricci curvature of the metric g(t). A important aspect in the treatment of the Ricci flow is the study of Ricci solitons, which generate self-similar solutions to the flow and often arise as singularity models. Given a semi-Riemannian manifold (M n , g), n ≥ 3, we say that (M, g) is a gradient Ricci soliton if there exists a differentiable function f : M −→ R (called the potential function) such that Ricg + Hessg ( f ) = λg,
λ ∈ R,
where Ricg is the Ricci tensor, Hessg ( f ) is the Hessian of f with respect to the metric g, and λ is a real number. We say that a gradient Ricci soliton is shrinking, steady, or expanding if λ > 0, λ = 0, or λ < 0, respectively. Bryant [2] proved that there exists a complete, steady, gradient Ricci soliton that is spherically symmetric for any n ≥ 3, which is known as Bryant’s soliton. In the bi-dimensional case an analogous nontrivial rotationally symmetric solution was obtained explicitly, and is known as the Hamilton cigar. Recently Cao and Chen [3] showed that any complete, steady, gradient Ricci soliton, locally conformally flat, up to homothety, is either flat or isometric to the Bryant’s R. Pina (B) · I. Menezes: IME - Universidade federal de Goiás, Caixa Postal 131, Goiânia, GO 74001-970, Brazil. e-mail: [email protected] I. Menezes: e-mail: [email protected] Mathematics Subject Classification: 53C21 · 53C50 · 53C25
https://doi.org/10.1007/s00229-019-01159-0
R. Pina, I. Menezes
soliton. The results obtained in [3] were extended to bach - flat gradient steady Ricci solitons (see [4]). Complete, conformally flat shrinking gradient solitons have been characterized as being quotients of Rn , Sn or R × Sn−1 (see [5]). In the case of steady gradient Ricci solitons, [6] provide all such solutions when the metric is conformal to an n-dimensional pseudo-Euclidean space and invariant under the action of an (n − 1)-dimensional translation group. Motivated by the notion of Ricci solitons on a semi-Riemannian manifold (M n , g), n ≥ 3, it is natural to consider geometric flows of the following type: ∂ g(t) = −2(Ric − ρ Rg) ∂t
(1)
for ρ ∈ R, ρ = 0, as in [7]. We call these the Ricci–Bourguignon flows. We notice that short time existence for the geometric flows described in (1) is provided in ([8]). Associated to the flows, we have the f
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