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 Iso(M,g) is large. More precisely we prove first that if dim(Iso(M,g)) ≥ 2dim(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/Iso(M,g)).