I have only learned about calculus and linear algebra, so I don't know about differential algebra.
I got to know about the concept of "geodesic" recently. What I need to know is this:
Suppose I have a $10\times 10$ matrix $Z$ and $Z = XY$, where $X$ is $10\times 3$, $Y$ is $3\times 10$. I have two points in a manifold, $(X_1, Y_1, Z_1)$ and $(X_2, Y_2, Z_2)$. I want to calculate the "geodesic path" between this two point in this manifold which satisfy $Z=XY$. How can I do that?
At present, I think the idea may be similar with how to calculate the geodesic path between two point $(x_1, y_1, z_2)$ and $(x_2, y_2, z_2)$ which satisfy $z=xy$ in a $3$-dimensional space. But I don't know how to calculate the simplified case easier, as I know nothing about differential geometry.
I'm looking forward to your response!