Ridigity of Ricci Solitons with Weakly Harmonic Weyl Tensors

2016/04/24 by Hwang, Seungsu, Yun, Gabjin
#53C25 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.07018

Abstract

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let (Mⁿ, g) be a compact gradient shrinking Ricci soliton satisfying \rm Ric_g + Ddf = ρg with ρ>0 constant. We show that if (M,g) satisfies δ\mathcal W (⋅, ⋅, ∇ f) = 0, then (M, g) is Einstein. Here \mathcal W denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in \citel-r, \citem-s and \citep-w3.

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