Extrinsic geometry of the Gromoll-Meyer sphere

Among a family of 2-parameter left invariant metrics on Sp(2), we determine which have nonnegative sectional curvatures and which are Einstein. On the quotient N˜¹¹=(Sp(2)×S⁴)/S³, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotient N˜¹¹/S³, we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere Σ⁷. Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature simultaneously.