I Tangent spaces at different points

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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

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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

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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

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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

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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.
 

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