# An example of a derivation at a point on a $C^k$-manifold which is not a tangent vector

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

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