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I am studying Thorpe's Elementary Topics in Differential Geometry, an undergrad level book. On this site, there are probably lots of technical answers for such a question, I am hoping for an answer that is commensurate with the level of this book. Very early on in the book (Chapter 2, Vector Fields), he introduces the notion of vectors attached to a point, and says vectors attached to different points cannot be added. **enter image description here**

Then a few pages later, he defines a vector field for the first time enter image description here

For the next few chapters I do not come across any notion of adding vectors attached to different points, but then, in Chapter 7, Geodesics, I come across this: enter image description here enter image description here

My question: Doesn't the very definition of the derivatives of $X_1(t),\dotsc $ here require the addition of vectors at neighbouring points to find the limit ? As in $X_1(t+ h)$ and $X_1(t)$ are attached at $ \alpha(t+h)$ and $\alpha(t)$, so how am I allowed to take the limit of $(X_1(t+h) - X_1(t))/h$ for $ h \longrightarrow 0$. I want to understand, how is it justified to write down derivatives of vector fields without allowing vector algebra of vectors attached to different points ?

If I were to be told, that $\mathbf{Y}(t)$ is a field defined at $\alpha(t)$ by the derivatives $\dfrac{d X_i(t)}{dt}$, I have no objection. Do I simply take $\dot{\mathbf{X}}(t) = \mathbf{Y}(t)$ as a definition for now, without trying to interpret $\dot{\mathbf{X}}(t)$ as a derivative of $\mathbf{X}(t)$ ?

The following questions are related, but they seem to be at a higher level. Resolving a contradiction: defining smooth vector fields if we cannot compare vectors at different tangent spaces?

Difference of vectors living in different tangent spaces

Tangent spaces, how are vectors parallel transported?

me10240
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1 Answers1

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I took a look at the book. Actually the author defines "velocity vector" of a curve on the very next page after the one you showed a picture of (right after he defines vector fields).

As some of the people in the comments above have already said, you can side-step the issue you mention. Since a curve is a function $\alpha \colon \Bbb{R} \to \Bbb{R}^n$, written as $\alpha(t) = (x_1(t), x_2(t), \dots, x_n(t))$, each individual component function $x_i(t)$ is just an ordinary single variable function $\Bbb{R} \to \Bbb{R}$. So you know what "derivative" $x_i'(t)$ means in that situation. The author simply makes it the definition of "velocity vector" that the components of $\dot{\alpha}(t)$ are $x_i'(t)$.

On the one hand, you can make sense of it as I (and the author, and the people in the comments) described above. But on the other hind, you are still right, this feels like "sweeping details under the rug", and that something technical is not being addressed.

Your question is actually a very good one, and this is a subtle and important point. There is something in geometry called a "connection" (sometimes "affine connection" or "linear connection"). It is some technical and systematic way of identifying tangent spaces $\Bbb{R}_p^n$ and $\Bbb{R}_q^n$ when $p$ and $q$ are "close", which allows adding and subtracting vectors in different nearby tangent spaces. I'm not sure if it's discussed in this book, but there is such an abstract concept in the background to make sense of this.

Nick
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