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In Lee's book 'Introduction to Smooth Manifolds', there is an interesting discussion (near the end of chapter three) of several different ways of viewing/constructing the notion of a tangent space to a point on a manifold. As it turns out, there are quite a few. Tangent vectors can be understood as:

  • derivations of the space of germs

  • equivalence classes of curves

  • tangent vectors to charts

  • equivalence classes of $n$-tuples

Lee makes the remark that the last of these, which is based on the transformation rule under coordinate transformations

$$\tilde v^j=\frac{\partial \tilde x^j}{\partial x^i}v^i$$

is actually the oldest way to make sense of the concept 'tangent vector', and is still heavily used by physicists. As someone who identifies as a physicist, I can confirm this ;-). However, the remark that this is the original way of understanding tangent spaces surprised me, and sparks the following question:

Who introduced the different constructions of tangent spaces (including the one in terms of the transformation law), when did they do this, and --- most importantly --- what was their motivation for taking the different approaches that we know now?

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    $\begingroup$ I am interested in the answer to your question. As to why we have different approaches the answer is fairly clear in books like Lee's. These are isomorphic pictures and it happens that proofs and calculations in one formalism are easier than others. Although, in the non-smooth category the space of coordinate derivations is not finite dimensional so, we're just talking about smooth manifolds here. Anyway, great question! $\endgroup$ – James S. Cook Nov 16 '14 at 19:04
  • $\begingroup$ It's a pity this question has not received an answer in four years. Have you tried asking it somewhere else, like MO? $\endgroup$ – Michael Bächtold Sep 19 '18 at 7:21
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I apologize for the limited information in this answer, but as the question has gone unanswered for 4 years, I plead that something is better than nothing.

Your fourth version appears in The Mathematical Theory of Relativity by Eddington, in a section titled "The mathematical notion of a vector". Eddington writes:

We have a set of four numbers $(A_1,A_2,$ $A_3,A_4)$ which we associate with some point $(x_1,x_2,x_3,x_4)$ and with a certain system of coordinates. We make a change of the coordinate system, and we ask, What will these numbers become in the new coordinates? The question is meaningless; they do not automatically "become" anything. Unless we interfere with them they stay as they were. But the mathematician may say, "When I am using the coordinates $x_1,x_2,x_3,x_4$, I want to talk about the numbers $A_1,A_2,A_3,A_4$; and when I am using $x'_1,x'_2,x'_3,x'_4$ I find that ... I shall want to talk about four different numbers $A'_1,A'_2,$ $A'_3,A'_4$. So for brevity I propose to call both sets of numbers by the same symbol $\mathfrak{A}$."

The mathematician soon realizes he cannot write down the coordinates for $\mathfrak{A}$ in every one of an infinite variety of coordinate systems, and continues:

"I see that I must alter my plan. I will give you a general rule to find the new values of $\mathfrak{A}$ when you pass from one coordinate system to another ..."

In mentioning a rule the mathematician gives up his arbitrary power... He binds himself down to some kind of regularity. Indeed we might have suspected that our orderly-minded friend would have some principle in his assignment of different meanings to $\mathfrak{A}$. Now Eddington asks, can we possibly know what kind of rule the mathematician would adopt, without knowing what problem he is working on? He answers his own question: I think we can; it is not necessary to know anything about the nature of his problem... it is sufficient that we know a little about the nature of a mathematician.

Using various assumptions, Eddington narrows the options until only the transformation laws for contravariant and covariant vectors remain.

Eddington then goes on to talk about the physicist's notion of a vector, but that's not relevant to your question.

Now some speculation. Older treatments that I've seen (i.e., early 20th C.) use the transformation definition. There's a famous quote by Weyl, usually given as "The introduction of numbers as coordinates is an act of violence" (see this question for the actual quote). It would not surprise me if Weyl looked for a coordinate-free definition of tangent vector.

I vaguely remember reading a passage by Weyl, in which he talked about the origin of the concept of a topological space. It went something like this. Differential manifolds are defined via patches with differentiable transition functions; Riemann surfaces via patches with conformal transition functions. What about continuous transition functions? And then the epiphany: in this case you can just axiomatize the patches, e.g., the intersection of any two patches is a patch. Replace the word "patch" with "open set" and you're good to go. (Of course we've widened the scope from topological manifolds to topological spaces in general.) Weyl then went on to lament that for differential manifolds and Riemann surfaces, no such "purely intrinsic" definition has ever been found.

I have not been able to locate this passage again. Also this version says nothing about Hausdorff's role, so I fear I must be misremembering some of the details.

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  • $\begingroup$ The idea behind the first version goes back in principle to Zariski, who used such a viewpoint in algebraic geometry. See mathoverflow.net/questions/287654. I do not know who carried the idea over from algebraic geometry to differential geometry. $\endgroup$ – KCd Nov 21 '18 at 13:25

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