First of all, I have a specific question. Suppose M is an m-dimensional Ck-manifold, for 1≤k<∞. Is the tangent space to a point defined as the space of Ck derivations on the germs of Ck functions near that point? If so, is it m-dimensional? Bredon's book Topology and Geometry comments that only in the C∞ 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 Ck-manifold would be bigger in the case k<∞. 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 Ck 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 Ck 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 Ck-manifolds; Hirsch's Differential Topology deals with them all throughout, and Duistermaat's book on Lie groups defines them as C2-manifolds. Should I, as a student of topology / geometry, be paying close attention to Ck-manifolds and the distinctions with the smooth case?
@Pedro: As you know, any Ck-atlas is compatible with a C∞-atlas. For Lie groups we have more: C1⟹Cω.
- 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∞(M)...Note that the Lie bracket of two vector fields is a vector field fails to be true if you restrict to C1 functions:
∂xi∂xjf=∂xj∂xif, if f∈C2(U)
- Differential topology deals with functions with less regularity (to use a most general form of Sard's theorem, Morse theory...)