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Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of derivations of the form $v: \mathscr{H}_x \to \mathbb{R}$. In his book on Lie groups and Lie algebras Serre proves that in case of analytic manifolds this embedding is indeed an isomorphism of vector spaces.

I was told that this is also true in case of $C^\infty$, but there are derivations which are not tangent vectors when $k < \infty$, and the definition of the tangent space (cf. here) is quite different, and these definitions are only shown to be equivalent when $k = \infty$.

So my questions are:
1) What is the correct way of defining the tangent space, after all? What breaks down if we try to use maximal ideals when $k < \infty$? An (obscured by adding $\mathbb{R}$) explanation is given here.
2) Can you please give me an explicit example of a derivation which is not a tangent vector? There is a proof of existence here, but no explicit example.

Also, a softer question: are $C^k$-manifolds ($k < \infty$) important in differential geometry or differential topology? E.g. in Dubrovin-Novikov-Fomenko's textbook the Sard's theorem is proved for $C^\infty$ and there was no mention of the case when $k < \infty$, this leads me to believe that $C^\infty$ and $C^\omega$ are the only important cases in differential topology. Is this really so?

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As per your last question, it's a theorem of Whitney (I'm not sure where it's published), that every $C^1$ manifold has a compatible $C^\infty$ atlas and that if two smooth manifolds are $C^1$-diffeomorphic, then they are $C^\infty$- diffeomorphic. That may partially explain why $C^k$ manifolds are far less common. – Jason DeVito Oct 18 '11 at 23:26
A tractation on the theorem Jason mentions can be found here: Munkres - Elementary Differential Topology – Giuseppe Negro Oct 20 '11 at 9:31
Have a look at Laird E. Taylor, The tangent space to a $C^k$ manifold, Bull. Amer. Math. Soc. 79 (1973), 746. – t.b. Oct 20 '11 at 15:38
@t.b., thanks, but this was proved in the Abraham et. al., albeit not as concisely. Is it too much to ask for an explicit example? :) – Alexei Averchenko Oct 20 '11 at 18:36
@Alexei: yes, it is too much to ask for, since you should figure that out yourself. My comment was intended for you to look at that proof and then build an example... :) How does a derivation given by a tangent vector look like in that proof? – t.b. Oct 20 '11 at 18:39
up vote 4 down vote accepted

Take a look at 294 and 295 in Manifolds. Tensor Analysis and Applications of Abraham, Marsden and Ratiu, cf. here.
The proposition 4.2.41 is the theorem of Newns and Walker which answers your question 2.

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Thanks, this book is wonderful! However, it proves the existence, but does not give any explicit example. At least I don't understand how to construct it from a given germ $|x|^{k + \varepsilon}$, $0 < \varepsilon < 1$. – Alexei Averchenko Oct 20 '11 at 9:13
@Alexei Yah, I like that book too. There is an apparent 3rd edition draft floating around on the internet (and it's been floating around for years). So, I'm wondering whether there are plans to release a new edition; has anyone heard about the future of this book? – ItsNotObvious Oct 20 '11 at 11:39

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