# I 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?

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#### Orodruin

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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

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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.

"Tangent spaces at different points"

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