In this paper we introduce the notion of generalized quasi--Einstein
manifold, that generalizes the concepts of Ricci soliton, Ricci almost soliton
and quasi--Einstein manifolds. We prove that a complete generalized
quasi--Einstein manifold with harmonic Weyl tensor and with zero radial Weyl
curvature, is locally a warped product with $(n-1)$--dimensional Einstein
fibers. In particular, this implies a local characterization for locally
conformally flat gradient Ricci almost solitons, similar to that proved for
gradient Ricci solitons
ABSTRACT. In this paper we consider a perturbation of the Ricci solitons equation proposed by J. P. Bourguignon in [23]. We show that these structures are more rigid then standard Ricci solitons. In particular, we prove that there is only one complete three-dimensional, positively curved, Riemannian manifold satisfyingfor some smooth function f . This solution is rotationally symmetric and asymptotically cylindrical and it represents the analogue of the Hamilton's cigar in dimension three. The key ingredient in the proof is the rectifiability of the potential function f . It turns out that this property holds also in the Lorentzian setting and for a more general class of structures which includes some gravitational theories.
In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons
In this paper we present some results on a family of geometric flows introduced by J. P. Bourguignon in [4] that generalize the Ricci flow. For suitable values of the scalar parameter involved in these flows, we prove short time existence and provide curvature estimates. We also state some results on the associated solitons.
Abstract. In this paper we prove that any complete conformal gradient soliton with nonnegative Ricci tensor is either isometric to a direct product R×N n−1 , or globally conformally equivalent to the Euclidean space R n or to the round sphere S n . In particular, we show that any complete, noncompact, gradient Yamabe-type soliton with positive Ricci tensor is rotationally symmetric, whenever the potential function is nonconstant.
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