Positively curved Riemannian metrics with logarithmic symmetry rank bounds
The author examines the relationship between the rational cohomology of a compact Riemannian manifold \((M,g)\) which admits a metric \(g\) of positive sectional curvature, and the rank of the isometry \(I(M,g)\). He proves Theorem: Let \(g\) be a metric of positive sectional curvature on a manifold \(M\) of dimension \(n\geq c\geq 2\). 1. Assume that \(\text{Rank}\{I(M,g)\}\geq 2\log_2(n)+7\). If \(M\) has the rational cohomology of a \(1\)-connected compact symmetric space \(N\), then \(N\) is a product of spheres times either a rank-1 symmetric space or a rank-\(p\) Grassmannian \(\mathrm{SO}(p+q)/\{\mathrm{SO}(p)\times\mathrm{SO}(q)\}\) with \(p\in\{2,3\}\). 2. If \(M\) is connected, closed, and \(\text{Rank}\{I(M,g)\}\geq2\log_2(n)+\frac c2-1\), then the Betti numbers \(\beta_{2i}(M)\) for \(2i<c\) agree with those of \(S^n\), \(\mathbb{CP}^{n/2}\), or \(\mathbb{HP}^{n/4}\). Furthermore, if \(n\equiv 0\mod 4\), then \(\beta_{2i+1}(M)=0\) for \(2i+1<c\). 3. If \(M\) is simply connected, closed and if \(\text{Rank}\{I(M,g)\}\geq 2\log_2(n)+\frac c2-1\), then \(H^*(M;\mathbb{Q})\) is 4-periodic up to degree \(c\), i.e., there exists \(x\in H^4(M;\mathbb{Q})\) so that the cup product with \(x\) is a surjection from \(H^i(M;\mathbb{Q})\) to \(H^{i+4}(M;\mathbb{Q})\) for \(0\leq i<c-4\) and an injection for \(0<i\leq c-4\). 4. No non-trivial connected sum with summands \(\mathbb{CP}^n\) and \(\mathbb{HP}^{n/2}\) admits a metric with positive sectional curvature so that \(\text{Rank}(I(M,g))\geq 2\log_2(4n)\). 5. If \(M^{4n+1}\) is a connected closed manifold which admits a metric of positive sectional curvature and if \(\text{Rank}(I(M,g))\geq 2\log_2(4n+1)\), then \(\pi_1(M)\) acts freely and isometrically on some positively curved rational homology \((4k+1)\) sphere and \(\pi_1(M)\approx\pi^\prime\times\pi^{\prime\prime}\) where \(\pi^\prime\) is cyclic with order a power of \(2\) and \(\pi^{\prime\prime}\) has odd order.
scientific article; zbMATH DE number 6383723
scientific article