Positively curved manifolds with symmetry

Abstract

There are very few examples of Riemannian manifolds with positive sectional curvature known. In fact in dimensions above $24$ all known examples are diffeomorphic to locally rank one symmetric spaces. We give a partial explanation of this phenomenon by showing that a positively curved, simply connected, compact manifold $(M,g)$ is up to homotopy given by a rank one symmetric space, provided that its isometry group $\mathrm{Iso}(M,g)$ is large. More precisely we prove first that if $\dim(\mathrm{Iso}(M,g))\ge 2\dim(M)-6$, then $M$ is tangentially homotopically equivalent to a rank one symmetric space or $M$ is homogeneous. Secondly, we show that in dimensions above $18(k+1)^2$ each $M$ is tangentially homotopically equivalent to a rank one symmetric space, where $k>0$ denotes the cohomogeneity, $k=\dim(M/\mathrm{Iso}(M,g))$.