- 780
- 36
How do you know if two given points on a manifold have the same tangent space? Checking if a vector does not change when transported from one point to the other is enough?
What do you mean by the same? They cannot be the same at different points. They are isomorphic, as e.g. on a n-manifold they both are isomorphic to ##\mathbb{R}^n##. So again, what is supposed to mean "equal"?How do you know if two given points on a manifold have the same tangent space? Checking if a vector does not change when transported from one point to the other is enough?
oh that makes senseThey cannot be the same at different points. They are isomorphic, as e.g. on a n-manifold they both are isomorphic to ##\mathbb{R}^n##
I mean, if we consider the m-dimensional manifold to be ##\mathbb{R}^m## itself, we can find a (global) coordinate system where the basis don't change, namely a Cartesian coordinate system. In that coordinate system, a vector (not a vector field, so constant components) will not change at all if we move from one point to another. That's what we usually do geometrically when we e.g. drag the arrows around in the plane, right?What do you mean by the same? So again, what is supposed to mean "equal"?
Just to add that this motivates the concept of connections, which, well, connect tangent spaces at different points. In euclidean n-space, the isomorphism is natural, but not so in general manifolds.oh that makes sense
I mean, if we consider the m-dimensional manifold to be ##\mathbb{R}^m## itself, we can find a (global) coordinate system where the basis don't change, namely a Cartesian coordinate system. In that coordinate system, a vector (not a vector field, so constant components) will not change at all if we move from one point to another. That's what we usually do geometrically when we e.g. drag the arrows around in the plane, right?
Now for me it seems unecessary to continue saying that the points in ##\mathbb{R}^m## have different tangent spaces, as we can do what I just described above. From this follows my question in post #1, if checking the constancy of a vector is a sufficient condition.