Cohomogeneity one actions on unit spheres

In this paper, we provide a complete classification for all the isometric cohomogeneity one actions on unit spheres. Using this theory, we can very easily classify all the isometric cohomogeneity one actions on the Riemannian symmetric spaces $\mathbb{C}{\mathrm{P}}^{m-1}$ and $\mathbb{H}{\mathrm{P}}^{m-1}$.