[This question](https://mathoverflow.net/questions/89345/example-of-a-manifold-which-is-not-a-homogeneous-space-of-any-lie-group) 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...)?



A followup on non-homogeneous spaces.
Yes, [Eschenburg constructed an infinite family of simply connected 7-dimensional examples](https://doi.org/10.1007/BF01389224 "Eschenburg, Jost-Hinrich. New examples of manifolds with strictly positive curvature. Invent. Math. 66, 469–480 (1982), EuDML:142893. zbMATH review at https://zbmath.org/0484.53031") and proved that many of them are not homotopy equivalent to homogeneous spaces. His examples are biquotients however. In fact the only known examples of closed positively curved manifolds are biquotients or spaces of cohomogeneity one and above dimension 13 only homogeneous examples (and space forms) are known.


