Munteanu, Ovidiu; Wang, Jiaping Positively curved shrinking Ricci solitons are compact. (English) Zbl 1405.53072 J. Differ. Geom. 106, No. 3, 499-505 (2017). A complete Riemannian manifold \((M,g)\) is a shrinking gradient Ricci soliton if the equation \[ \text{Ric}+\text{Hess}(f)=\frac12 g \] holds for some function \(f\). In the work [arXiv e-print service , Paper No. 0303109, 22 p. (2003; Zbl 1130.53002)], G. Perelman proved that if \((M,g)\) is a three-dimensional non-collapsing gradient shrinking Ricci soliton with positive and bounded sectional curvature, then \((M,g)\) must be compact.In [Chin. Ann. Math., Ser. B 27, No. 2, 121–142 (2006; Zbl 1102.53025)], H.-D. Cao posed the question whether this theorem is true in full generality in any dimension.In the present paper the authors prove the following:{Theorem}. Let \((M,g)\) be an \(n\)-dimensional gradient shrinking Ricci soliton with nonnegative sectional curvature and positive Ricci curvature. Then \((M,g)\) must be compact.By combining the above theorem with results of O. Munteanu and N. Sesum [J. Geom. Anal. 23, No. 2, 539–561 (2013; Zbl 1275.53061)] and P. Petersen and W. Wylie [Geom. Topol. 14, No. 4, 2277–2300 (2010; Zbl 1202.53049)], they obtain the following:{Corollary}. Let \((M,g)\) be an \(n\)-dimensional gradient shrinking Ricci soliton with nonnegative sectional curvature. Then \((M,g)\) must be compact or a quotient of \(\mathbb{R}^n\) or of the product \(\mathbb{R}^k\times N^{n-k}\) with \(1\leq k\leq n-2\), where \(N\) is a compact simply connected shrinking Ricci soliton of dimension \(n-k\) with positive Ricci curvature. Reviewer: Andreas Arvanitoyeorgos (Patras) Cited in 1 ReviewCited in 41 Documents MSC: 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) Keywords:Ricci soliton; Ricci curvature; sectional curvature Citations:Zbl 1130.53002; Zbl 1102.53025; Zbl 1275.53061; Zbl 1202.53049 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid