Extrinsic geometry of the Gromoll-Meyer sphere

Abstract

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

22 pages,to appear in Differential Geometry and its Applications