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 ${\tilde{N}}^{11}=(Sp(2)\times S^4)/S^3$, we construct a homogeneous isoparametric foliation with isoparametric hypersurfaces diffeomorphic to $Sp(2)$. Furthermore, on the quotient ${\tilde{N}}^{11}/S^3$, 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.

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Gromoll-Meyer球的外在几何形状

在2参数左不变指标族中 $\mathrm{小号}p（2）$，我们确定哪些具有非负截面曲率，哪些是爱因斯坦。关于商${\stackrel{〜}{ñ}}^{11}=（\mathrm{小号}p（2）\times {\mathrm{小号}}^4）/{\mathrm{小号}}^3$，我们构造了一个等参的等参叶面，其中等参超曲面对 $\mathrm{小号}p（2）$。此外，关于商${\stackrel{〜}{ñ}}^{11}/{\mathrm{小号}}^3$，我们构造了一个具有向Gromoll-Meyer球面变态的超正规超曲面的超正规系统 ${\mathrm{\Sigma }}^7$。此外，每个超表面上的诱导度量同时具有正Ricci曲率和准正截面曲率。