A Minimal Brieskorn 5-sphere in the Gromoll-meyer Sphere and Its Applications

We recognize the Gromoll-Meyer sphere Σ as the geodesic join of a simple closed geodesic and a minimal subsphere Σ ⊂ Σ, which can be equivariantly identified with the Brieskorn sphere W 5 3 . As applications we in particular determine the full isometry group of Σ, classify all closed subgroups which act freely, determine the homotopy types of the corresponding orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the Calabi involution of W 5 3 , and obtain explicit formulas for diffeomorphisms between the two Brieskorn spheres W 5 3 and W 13 3 and standard Euclidean spheres.