We show that the only compact positively curved Riemannian manifolds of even dimension acted on by a simple group with a codimension one orbit are the compact rank one symmetric spaces.
"Cohomogeneity One Manifolds of Even Dimension with Strictly Positive Sectional Curvature." J. Differential Geom. 68 (1) 31 - 72, September 2004. https://doi.org/10.4310/jdg/1102536709