In the Stiefel manifold $X_{n,k}$, we replace Frankel linear height function by a quadratic one. We prove this is still a Morse-Bott function, whose structure of critical levels presents a dichotomy according to the sign of $n-2k$. The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

中文翻译：

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四元数 Stiefel 流形上的非线性 Morse-Bott 函数

在 Stiefel 流形 $X_{n,k}$ 中，我们用二次函数代替 Frankel 线性高度函数。我们证明这仍然是一个 Morse-Bott 函数，其临界水平结构根据 $n-2k$ 的符号呈现出二分法。临界子流形不再是格拉斯曼流形，而是两个格拉斯曼流形的乘积的基纤维化的总空间。我们明确地整合了梯度流。