Tangent spaces at different points

[kent davidge]

How do you know whether two points $p$ and $q$ of a manifold have the same tangent space?

If the two tangent spaces are equal, then the vectors in the two tangent spaces are exactly the same. I suspect that it's equivalent to picking a vector at $p$ and dragging it to $q$ and the vector will not change. So perhaps a test to check whether the tangent spaces are the same is to see if the covariant derivatives of any vector at $p$ w.r.t. a vector field which generates flow which maps the point $q$ vanish.

Is this a good test?

 

[Orodruin]

How do you know whether two points $p$ and $q$ of a manifold have the same tangent space?

That is easy, they don't.

 

[PeterDonis]

If the two tangent spaces are equal

You are misunderstanding how tangent spaces work. The tangent space at every point is a different space. In order to compare vectors in the tangent spaces at two different points, you need what is called a connection. With a connection, you can meaningfullly test whether a vector in the tangent space at $p$ is the same as, or different from, a vector in the tangent space at $q$. But even with a connection, the tangent spaces themselves are never "the same"; it's meaningless even to ask that. They're distinct spaces.

 

[fresh_42]

How do you know whether two points $p$ and $q$ of a manifold have the same tangent space?

A tangent space is the pair $(p,T_pM)$, so even if $T_pM \cong T_qM$ which they usually are, since an n-manifold has tangent spaces which are all isomorphic to $\mathbb{R}^n$, the tangent spaces themselves are different, because the points $p$ and $q$ are different and the tangents are fixed at these points. A tangent on a parabola is always a straight, and as a vector space isomorphic to $\mathbb{R}$. Nevertheless are those straights different at different points. And even if we had the same slope at many points, as e.g. on a sine function, they are still different, because their own coordinate systems have different origins. A connection is a possibility to move one to the other. However, this is in general not unique, because it depends on how the transport is done.

 

[WWGD]

And even if we had the same slope at many points, as e.g. on a sine function, they are still different, because their own coordinate systems have different origins. A connection is a possibility to move one to the other. However, this is in general not unique, because it depends on how the transport is done.

Still, I guess the Levi-Civita connection is distinguished , in some respects, albeit not canonical.

 

[Orodruin]

Still, I guess the Levi-Civita connection is distinguished , in some respects, albeit not canonical.

He is not referring to the transport being connection dependent (which is rather clear). He is referring to it being generally path dependent.