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First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the germs of $C^k$ functions near that point? If so, is it $m$-dimensional? Bredon's book Topology and Geometry comments that (p.77) only in the $C^\infty$ case can one prove that every derivation is given by a tangent vector to a curve. If so, this would suggest that (if indeed given this definition), the tangent space to a $C^k$-manifold would be bigger in the case $k < \infty$. Additionally, out of curiosity, would anybody have an example of a derivation that is not a tangent vector to a curve?

Secondly, it would seem to me that a fair share of the things I learned about smooth manifolds should fail or at least require more elaborate proofs in the $C^k$ case. We only used higher derivatives in proving Sard's theorem, but all the time we used the identification that the tangent space is given by tangent vectors to curves; the tubular neighborhood theorem comes to mind. What are the standard facts of smooth manifolds that do fail in the $C^k$ case?

Thirdly, are they really important? It seems a lot of books deal only with smooth manifolds, but a fair share also seem to deal with $C^k$-manifolds; Hirsch's Differential Topology deals with them all throughout, and Duistermaat & Kolk's book Lie groups (p.1) defines them as $C^2$-manifolds. Should I, as a student of topology / geometry, be paying close attention to $C^k$-manifolds and the distinctions with the smooth case?

Alp Uzman
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Pedro
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    I remember losing a little bit of sleep about exactly those questions. My understanding is that every maximal $C^k (k\geq 1)$ atlas contains a $C^\infty$ atlas, which is a relief. – Tim kinsella Aug 31 '14 at 11:07
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    I'll just say that yes it is indeed a problem to define the tangent space via the derivations on a $C^k$ manifold and the space generated in such a way is indeed infinite dimensional. But I have no reference. On the other hand as said by @Timkinsella you can always get a $C^\infty$ atlas from a $C^k$ atlas. – Lolman Aug 31 '14 at 11:13
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    Duistermaat defines Lie groups as $C^2$-manifolds because that suffices to prove they are analytic (better than $C^{\infty}$). Nice exposition at http://terrytao.wordpress.com/2011/06/21/the-c11-baker-campbell-hausdorff-formula/ – Colin McLarty Aug 31 '14 at 11:20
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    @Lolman: Very interesting! How is it defined then? It'd be lovely if someone could get a reference for showing it's infinite-dimensional in the derivations definition. – Pedro Aug 31 '14 at 11:27
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    You use a more basic idea and differentiate along differentiable curves! See here: http://en.wikipedia.org/wiki/Tangent_space#Definition_as_velocities_of_curves For the proof look at this: http://math.stackexchange.com/questions/537528/whats-special-about-c-infty-functions Basically the best idea is to look at the definition of the dual of the tangent space on some $P$, and see that if $k<\infty$ then not all $g\in C^k$ vanishing at $P$ and with vanishing first order derivatives will be in the square of the ideal of all the $C^k$ functions vanishing at $P$. – Lolman Aug 31 '14 at 11:46
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    Much of this question is answered at http://math.stackexchange.com/questions/73677/an-example-of-a-derivation-at-a-point-on-a-ck-manifold-which-is-not-a-tangent – Colin McLarty Aug 31 '14 at 11:57
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    As a general comment, you shouldn't use the word "tangent space" to refer to the space of all derivations. Even in the $C^k$ context, the word "tangent space" means the (finite-dimensional) space of all tangent vectors. Books that restrict to the $C^\infty$ case sometimes define the tangent space using derivations, but that's not the primary definition. In particular, books that consider the $C^k$ case generally ignore derivations completely in favor of some other formalism. – Jim Belk Jun 04 '15 at 16:48
  • Also relevant to the matter of derivations on $C^k$ functions is Exr. 18 on pp.124-125 of Jeffrey M. Lee's book "Manifolds and Differential Geometry". – Alp Uzman May 14 '22 at 00:43

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@Pedro: As you know, any $C^k$-atlas is compatible with a $C^\infty$-atlas. For Lie groups we have more: $C^1\Longrightarrow C^\omega$.

  1. Differential geometry deals only with smooth atlas (in order to identify a tangent vector with a derivation, to work with vector fields as derivations on the algebra of functions $C^\infty(M)$...Note that the Lie bracket of two vector fields is a vector field fails to be true if you restrict to $C^1$ functions: $$\partial_{x_i}\partial_{x_j}f=\partial_{x_j}\partial_{x_i}f,\ \text{if}\ f\in C^2(U)$$
  2. Differential topology deals with functions with less regularity (to use a most general form of Sard's theorem, Morse theory...)
amine
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    The existence of a smooth structure may not be much help. The level set of $C^k$ function in Euclidean space cannot be given a smooth structure that is compatible with the smooth structure of Euclidean space. – shuhalo Oct 24 '17 at 04:56