Positively curved manifolds with low fixed point cohomogeneity (Q2494183)

Let \(G\) be a connected, compact Lie group of isometries acting on a closed, Riemannian manifold \(M\). The cohomogeneity of the action, \({\text{cohom}}\,(M, G)\), is, by definition, the dimension of the orbit space \(M/G\). It gives a natural measure of the size of \(G\). Note that \({\text{cohom}}\,(M, G)=0\) means that \(G\) acts transitively on \(M\), hence \(M\) is homogeneous and the fixed point set is \(M^G = \emptyset\). If \(M^G \neq \emptyset\), its dimension constrains the cohomogeneity, thus the finer measurement \({\text{cohomfix}}\,(M, G)= \dim M/G - \dim M^G -1\) introduced by the first author a few years ago, becomes more natural. Since for \(M^G = \emptyset\), one has \({\text{cohom}}\,(M, G)= {\text{cohomfix}}\, (M, G)\), the latter is clearly an extension of the first. Throughout this paper, it is assumed that \(M^G \neq \emptyset\). Under this assumption, the authors classify fixed point cohomogeneity one, simply connected manifolds with positive curvature by proving that each one is equivariantly diffeomorphic to a compact, rank one symmetric space. Thus, the paper is a new step toward the classification of positively curved manifolds with large isometry groups.