If I have two matrices $A(0)$ at $t=0$ and $A(1)$ at $t=1$, they are $N\times N$ matrices, and they are on the Riemannian manifold of rank $K$. How to calculate the geodesic path $A(t)$? I haven't learned about Riemannian geometry, but I learned some concept of differential geometry from book. Anyone knows how to calculate it or know some materials discussing this problem? Looking forward to your reply!
Update: http://imajna.oxfordjournals.org/content/33/2/481.full.pdf?keytype=ref&ijkey=Tg3KpYcTdBwMjaJ I found this paper describes how to calculate the geodesic path given $A(0)$ and the tangent vector, which is got by solve a IVP. However, it didn't tell us how to solve the problem I state here, which should solve a BVP. The paper said that it is out of its scope.
If a closed form is not available, how to solve it numerically?