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 quotiente ${\tilde{N}}^{11}=(Sp(2)\times S^4)\mathrm{/}{S}^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to Sp(2). Furthermore, on the quotiente ${\tilde{N}}^{11}\mathrm{/}{S}^3$, we construct a transnormal system with transnormal hypersurfaces diffeomorphic to the Gromoll-Meyer sphere ${\mathrm{\Sigma }}^7$. Moreover, the induced metric on each hypersurface has positive Ricci curvature and quasi-positive sectional curvature simultaneously.