Define a metric on Stiefel manifold $V_{n,p}$ as $$\left<\Delta_1,\Delta_2\right>=\text{tr}\Delta_1^T\left(I-\frac{1}{2}YY^T\right)\Delta_2$$ $\forall \Delta_1,\Delta_2\in T_YV_{n,p}$
- how to calculate geodesic through the variation problem
$$\min\limits_{Y(t)}\int\left<\dot{Y},\dot{Y}\right>^{\frac{1}{2}}dt$$ $Y(t)$ is the curve in $V_{n,p}$.
Any advice is helpful. Thank you.