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I understand that tangent vectors lie in separate tangent spaces based on the point on which they are tangent to a manifold, but what about vectors that are parallel transported?

For any manifold $M$, there are an infinite number of tangent spaces $T_xM$ which are defined based on the point $x$ at which the space is tangent to the manifold. I know that if a vector field is defined on a manifold, then a vector $\vec{v}$ at a point $p$ must satisfy $\vec{v} \in T_pM$. This all makes sense to me. However, I just can't wrap my head around how vectors can be parallel transported around manifolds. Wouldn't that upset the overall structure of all vectors belonging to the tangent space of a particular point?

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    $\begingroup$ Here is my understanding. A manifold structure itself does not possess the ability to do parallel transport. As you said, a tangent space is local in its nature. It is only after some geometrical structure is imposed on the manifold (for parallel transport, the connection) that we can transport a vector from one tangent space to another. Note that a connection is a global structure. One illustration is the Chern number, defined using connection, can be used to characterize the topology(global properties). Since I'm not an expert in geometry, let's wait for some more comprehensive answer^_^ $\endgroup$ – Zheng Liu Nov 23 '15 at 6:36
  • $\begingroup$ The tangent bundle contains all pairs $(p, v)$ where $p\in M$ and $v\in T_p$. $\endgroup$ – John Douma Nov 23 '15 at 6:36
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    $\begingroup$ A parallel transport map is just a linear map between two tangent spaces to $M$ at endpoints of a path in $M$. A given vector $v$ in $T_{x}M$ does not itself "move" to $T_{y}M$. The linear map simply maps $v$ on a different vector $T_{y}M$. $\endgroup$ – Sinister Cutlass Nov 23 '15 at 6:37
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While the tangent spaces $T_pM$ are all distinct abstract vector spaces, if $M$ has dimension $n$, then they are all isomorphic to $\mathbb{R}^n$, even though there is a priori no "canonical" isomorphism $T_p M \cong \mathbb{R}^n$. On a Riemannian manifold, parallel transport gives a way of associating to a curve $\gamma(t) \subset M$ in the manifold isomorphisms of all the tangent spaces along the curve:$$f_t: T_{\gamma(0)}M \to T_{\gamma(t)}M.$$Thus, a vector in the tangent space at the point $\gamma(0)$ can be parallel transported to a vector $f_t(v)$ in the tangent space at $\gamma(t)$.

Note that this way of identifying two tangent spaces at different points is very noncanonical. First, it depends on the choice of a Riemannian metric. Second, it may depend on the particular path chosen to join the points.

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