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Terry Tao defines tangent space here as equivalence classes of continuously differentiable curves $\gamma : I \rightarrow G$ where $I$ is an open interval.

On the other hand, Wikipedia defines it as equivalence classes of curves $\gamma : (-1,1) \rightarrow M$.

I have 2 questions regarding this definition:

(i) Why does it have to be functions from open sets? Why can it not be paths $\gamma : [0,1] \rightarrow G$?

(ii) Are these definitions really equivalent?

The context in which I am asking this is Lie groups and Lie algebras which I just started to read about. Many thanks for your help.

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(i) I'm not certain, but it feels to me like differential geometry (and perhaps other areas of math) uses open neighborhoods almost reflexively. In any case, (ii) yes, though simple reparametrization. –  anon Sep 19 '11 at 10:51
@anon: (ii) I think the question was more about dropping the condition of continuous differentiability, less about the trivial change from a general open interval to $(-1,1)$. –  joriki Sep 19 '11 at 11:04
@joriki: That's not how I read it, but I suppose we should wait for OP. –  anon Sep 19 '11 at 12:00
Thanks, actually anon's read it right. I'm just generally confused at the moment by the new subject of lie algebras and groups and I didn't see that it was so obvious. –  Matt N. Sep 19 '11 at 13:09
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1 Answer

up vote 11 down vote accepted

(Let the usual symbols stand for the usual objects: $M$ a smooth manifold, $\gamma$ a smooth curve, etc.)

(1) One reflexively uses open sets because it's easiest to think of the tangent space at $x\in M$ as the set of velocities of curves passing through $x$. For this, you need to be able to differentiate a curve at $x$, and to differentiate, you need an open neighborhood about the (a) preimage of $x$.

So you could certainly define the curves on a closed interval as long as the interval has interior and the (a) preimage of $x$ is in the interior of the interval. It's just neater to define the tangent space with open intervals, because you're going to be working with the interior of the closed interval anyway.

(2) Yes. I'm going to leave this as an exercise for you (commenter anon suggested the idea). In fact, there are several other definitions for you to become familiar with:

$T_xM$ is ...

  • Equivalence classes of curves from an open interval (or the standard open interval) to $M$ passing through $x$;
  • Equivalence classes of derivations of functions at $x$;
  • Equivalence classes of pullbacks of tangent vectors in $\mathbb{R}^n$ via chart maps;
  • The fiber over $x$ in the tangent bundle (which is then defined as the appropriate quotient of a disjoint union of $\mathbb{R}^n\times $(chart neighborhoods) ).

You should check that these are all equivalent to the Wikipedia/Tao definition. This is just a long definition chase, so I don't feel compelled to outline details.

For more information, check out any introductory graduate differential topology textbook. I can recommend Riemannian Geometry by Manfredo do Carmo, Semi-Riemannian Geometry with Applications to Relativity by Barrett O'Neill, and of course for a technically beautiful but less readable introduction there's always Introduction to Differentiable Manifolds and Lie Groups by Frank Warner.

PS- You mentioned you're thinking about this in the context of Lie groups. Bonus points for seeing how $\mathfrak{g}$ is both $T_eG$ and the space of left-invariant vector fields.

(Edit: Added titles and full author names at request.)

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Thanks! And thanks for the book recommendation. As for the bonus: I'm still struggling with definitions so I'm not quite ready to answer that. –  Matt N. Sep 19 '11 at 13:06
+1 for the answer. I would even go to +2 (so to speak) if and when the references will be completed (simply the titles of the books and full author's names seem to do the job). Some readers are not as familiar with the literature as others and this should help them greatly. –  Did Sep 19 '11 at 13:40
@Surly Nice answer, but I must respectfully disagree that the equivalence of definitions of the tangent space is a a trivial "definition chase" as you suggest, at least for a beginner (such as myself). Jeffery Lee's "Manifolds and Differential Geometry", for instance, spends about the first 10-15 pages of chapter 2 covering aspects of this topic –  ItsNotObvious Sep 19 '11 at 17:16
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