  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.
