Let's say we have two $d$ dimensional subspaces with principal angles $\theta_1, \dots, \theta_d$. Then, if $U,V$ are the orthonormal bases for these two subspaces, the singular values of $U'V$ are $\cos\theta_1,\dots,\cos\theta_d$. Now we define $$D(U,V) = \sqrt{\det(I-U'VV'U)}= \prod_{i=1}^d \sin\theta_i$$ My question is: is the above defined $D(U,V)$ a semi-metric? Essentially, I am asking, does the triangle inequality hold here, i.e., $D(U,W)\leq D(U,V)+D(V,W)$?
There is a similar post: Does the product of principal sines between subspaces satisfy the triangle inequalilty?
But no explicit answer was given there.