Resumen de Positively curved manifolds with symmetry

Burkhard Wilking

• There are very few examples of Riemannian manifolds with positive sectionalcurvature known. In fact in dimensions above 24 all known examplesare diffeomorphic to locally rank one symmetric spaces. We give a partialexplanation of this phenomenon by showing that a positively curved, simplyconnected, compact manifold (M,g) is up to homotopy given by a rank onesymmetric space, provided that its isometry group Iso(M,g) is large. Moreprecisely we prove first that if dim(Iso(M,g)) ¡Ý 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/Iso(M,g)).