Abstract: Closed Riemannian manifolds with positive sectional curvature are a class of fundamental objects in Riemannian geometry. There are few known examples of such manifolds, each of which possesses interesting geometric properties. For example, they usually have large symmetry groups. In this talk, I will review a few classical results in positive curvature, including Bonnet-Myers theorem and Synge theorem. Then I will move to the special case of dim 6, talking about the classification problem of positively curved 6-manifolds with certain non-Abelian symmetry conditions. I showed that such manifolds have Euler characteristic 2,4, or 6, and obtained topological classification after imposing certain restrictions on the action.
