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?
Best Answer
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.