Stability of the \(p\)-curvature positivity under surgeries and manifolds with positive Einstein tensor
The class of manifolds admitting a metric of positive scalar curvature is closed under surgery in codimension at least 3, a result due to \textit{M. Gromov} and \textit{H. B. Lawson} [Ann. Math., II. Ser. 111, 423-434 (1980; Zbl 0463.53025)] and \textit{R. Schoen} and \textit{S.-T. Yau} [Manuscr. Math. 28, 159-183 (1979; Zbl 0423.53032); Publ. Math. Res. Cent. Univ. Wis. Madison 54, 235-242 (1987; Zbl 0658.53040)]. The author carries this over to the case of manifolds with positive \(p\)-curvature under the assumption that the surgery is in codimension at least \(p+3\). \(p\)-curvature is the average of all sectional curvatures orthogonal to a given \(p\)-plane. 0-curvature is nothing but scalar curvature, 1-curvature is the quadratic form associated with the Einstein tensor. As a consequence, the author obtains existence results for metrics with positive Einstein tensor on certain classes of \(n\)-manifolds, \(n\geq 7\), in particular on any compact 2-connected 7-manifold.
scientific article; zbMATH DE number 1080030