# Calculate Geodesic Path of $N\times N$ matrix on Riemannian manifold of fixed rank

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?

• What is "The Riemannian manifold of rank K"? Do you have in mind the standard definition of rank in terms of parallel Jacobi fields? Then there is no "the" manifold of rank k, there are infinitely many. – Moishe Kohan Jun 16 '14 at 18:29
• Hi @studiosus , for example, N = 100, and k = 10. I mean the matrices are of fixed rank, and to make it general, I use k, because I need to try different values of k. – Excalibur Jun 16 '14 at 21:50
• Then you have to explain what Riemannian metric you are using. Is it the restriction of the Euclidean metric on the space of all $N\times n$ matrices? Then I very much doubt that there is a reasonably "nice" formula. – Moishe Kohan Jun 16 '14 at 22:24
• @studiosus Actually I am not familiar with Riemannian Geometry, that is why it is hard for me to find a solution through papers. I am not sure about restriction of the Riemannian metric, so I think we can suppose no restriction. Sorry if I said something stupid! – Excalibur Jun 17 '14 at 0:08
• What studiosus is getting at is that you need to define what riemannian metric (RM) you want to consider: a manifold, such as the manifold of matrices of given rank, may admit many different RMs, and usually different metrics will define different geodesics. About the induced metric, if $S$ is a submanifold of a manifold $M$ with a RM, then the RM of $M$ induces a RM on $S$ called the induced metric. – Olivier Bégassat Jun 17 '14 at 0:32