Of course, this makes sense, thank you!
On a side (or maybe main)-note, do you know if there is some other quantity that uniquely defines the manifold?
In my mind, if you start deforming a sphere, you can end up with infinite unique shapes, but the deviation from the original curvature can be calculated by the curvature tensor field.
Is there some way to define the manifold without having to calculate the curvature tensor all over the manifold? (I want to do it numerically)
I am sorry if I am asking something that makes no sense, but as I said, this is pretty new to me :)