Positively curved manifolds with almost maximal symmetry rank
The author is interested in the classification, up to homeomorphism, of those positively curved closed manifolds \((M^n,g)\) whose isometry groups are large, in the sense that the rank of these isometry groups is high. This rank is called the \textit{symmetry rank} of the manifold. In [J. Pure Appl.\ Algebra 91, 137-142 (1994; Zbl 0793.53040)], \textit{K.~Grove} and \textit{C. Searle} showed that the symmetry rank can at most be equal to \(\lfloor (n+1)/2\rfloor\), where \(n\)~is the dimension of the manifold. Moreover, equality occurs only for spaces diffeomorphic to a sphere, a real or complex projective space, or a lens space. Here, the author studies the topology of positively curved manifolds with \textit{almost} maximal symmetry rank \(\lfloor (n-1)/2\rfloor\). He shows that a simply connected positively curved manifold \((M^5,g)\) with symmetry rank~\(2\) is homeomorphic to a \(5\)-sphere and provides information on the Euler characteristic and on the rational homotopy groups in a number of cases. He also investigates in how far his results extend to positively curved manifolds with only suitably large symmetry rank 0.
scientific article; zbMATH DE number 1879691
scientific article