[This question][1] asks for an example of a manifold which is not a homogeneous space of any Lie Group, and many examples are given in the answers. However: is there a an example known with a metric of positive sectional curvature (this, apparently, is a question asked by Marcel Berger many years ago, so was open then, but I have no idea of its current status...)?


  [1]: http://mathoverflow.net/questions/89345/example-of-a-manifold-which-is-not-a-homogeneous-space-of-any-lie-group