Circle actions and scalar curvature (Q2787995)

The existence of metrics of positive scalar curvature on some manifolds is considered. The main results are the following:NEWLINENEWLINE Theorem I. Let \(G\) be a compact Lie groups. Assume that there is a circle subgroup \(S^1\subset Z(G)\) contained in the center of the group \(G\). Let \(M\) be a closed connected effective \(G\)-manifold such that there is a component \(F\) of \(M^{S^1_0}\) with \(\text{codim\,}F= 2\). Then there is a \(G\)-invariant metric of positive scalar curvature on the manifold \(M\).NEWLINENEWLINE Theorem II. Let \(M\) be a closed simply connected semi-free \(S^1\)-manifold of dimension \(n>5\). If \(M\) is not spin or spin and the \(S^1\)-action is odd, then the equivariant connected sum of two copies of the manifold \(M\) admits an invariant of positive scalar curvature.NEWLINENEWLINE We mention an elegant statement that every torus manifold admits an invariant metric of positive scalar curvature.