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I have a question about tangent vectors of manifolds.

Imagine that I have a vector $V$ living in $T_pM$ and $W$ in $T_qM$.

In my book it is written that the difference between those vectors is ill defined.

I would like to really understand why.

Indeed If my manifold has dimension $m$, $V$ and $W$ are vectors of same dimension so I could imagine to subtract them.

I understood that it is because for example if I have the coordinates of $V$ in a given basis in $T_pM$ I would have no idea of the coordinates $V$ would have in $T_qM$ (because : how to associate a basis of $T_pM$ to a basis of $T_qM$).

But if I take $M=\mathbb{R}^n$, we can compare vectors of different points. So what makes it work in $\mathbb{R}^n$ and not in any general manifold $M$ (we have no problem of associating basis here for example).

I think that an answer to this last question would help me to visualise better things.

PS : I'm a beginner in differential geometry so not too complex answers please :)

levap
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StarBucK
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1 Answers1

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To see why this is ill-defined in general, think about the simplest non-trivial manifold such as the sphere $S^2$ in $\mathbb{R}^3$. At different points $p,q \in S^2$ you have generally different tangent spaces as in the image below:

enter image description here

and so it makes no sense to subtract the tangent vectors. Even if you try and subtract them inside $\mathbb{R}^3$ which makes sense, you won't get a vector that will belong to the tangent space at $p$ or $q$.

The reason everything works in $\mathbb{R}^n$ is that you have a natural notion of parallel transport which allows you to identify tangent vectors at different points using translations in a path-independent way. If you have a vector $v$ whose "origin" is at a point $p \in \mathbb{R}^n$ you can translate it to any other point $q \in \mathbb{R}^n$ so that it will start at $q$ by dragging it from $p$ to $q$ leaving the vector parallel all the time. This does not make immediate sense for general manifolds and leads to the notion of a connection.

levap
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