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If $M$ is a smooth manifold and $TM$ is the tangent bundle clearly $T_pM\cong T_qM$ (as vector spaces) for every $p,q\in M$. Nobody ensures that the previous vector spaces isomorphism is natural (or canonical). In $\mathbb R^n$ we have that $T_p\mathbb R^n$ and $T_q\mathbb R^n$ are naturally isomorphic to $\mathbb R^n$ so we can differentiate a vector field along a direction, in the usual way so taking the directional derivatives of each component.

If the isomorphism between tangent spaces in different point isn't natural, why can't we differentiate a vector field in the usual way? The problem is comparing vectors belonging in different (isomorphic) vector spaces; but we can send the two vectors, with an isomorphism, in a common vector space and then subtract them. Where is the importance of a natural isomorphism?

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    If you change what you mean by "with an isomorphism in a common vector space" then your result pulled back to the original tangent spaces will change. That is, what you are proposing will depend on *how* you make your two tangent spaces isomorphic to a common vector space. There are lots of isomorphisms between two vector spaces of the same dimension.2012-11-19
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    It is true that there are lots of isomorphisms between vector spaces, but there are also lots of natural isomorphisms. For example canonical automorphisms of $V$ are the elements of $Z(GL(V))$, so homothetic transformations.2012-11-20
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    So even if I choose a natural isomorphism, the directional derivative will not depends on the basis imposed on tangent spaces, but it depends on the choice of the natural isomorphism.2012-11-20
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    There is no such thing as a natural isomorphism between two DIFFERENT vector spaces except in trivial cases.2012-11-20
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    So is there only a natural isomorphism between two different vector spaces? Why?2012-11-20
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    What Kofi says is not quite right (e.g., a finite-dimensional vector space and its double dual are naturally isomorphic), but I suppose what Kofi means is that two vector spaces of the same dimension are generally not isomorphic in a canonical way. Galoisfan, do you think tangent spaces at different points of a general (abstract) manifold are naturally isomorphic in some way? What does the term "natural isomorphism" mean to you?2012-11-20
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    I think that in general two tangent spaces are not isomorphic in canonical way. Previously you said that there are lots of (not canonical) isomorphism between vector spaces, so you suggest that using canonical isomorphisms we have a unique way to identify the two vector spaces.2012-11-20
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    My objection is the following: if we prove that there are lots of canonical isomorphisms between two fixed vector spaces, crearly the unicity of the identification will fail.2012-11-20
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    There is no such thing as a canonical isomorphism in general between two vector spaces and it is hopeless to prove there are "lots" of them between two fixed vector spaces. As an analogy, there is no such thing as a natural inner product on a (finite-dim.) real vector space. Let $V$ be the vector space of real polynomials $a+bx+cx^2$ that vanish at 45. There are many inner products on $V$, but none is "natural" or "canonical", right?2012-11-21
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    Ok, it's clear now.2012-11-21

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