I saw three different definition of tangent space on wikipedia#tangent-space:
- via equivlent class of tangent curves: It works for both $C^k$ manifolds and smooth manifolds
- via derivations: It only works for smooth manifolds
- via cotangent space
I am curious abount if the third definition works for $C^k$ manifolds. On wikipedia the context is about smooth manifolds. I quote it here:
Again, we start with a $C^{\infty }$ manifold $M$ and a point $x\in M$. Consider the ideal $I$ of $C^{\infty }(M)$ that consists of all smooth functions $f$ vanishing at $x$, i.e., $f(x)=0$. Then $I$ and $I^{2}$ are both real vector spaces, and the quotient space $I/I^{2}$ can be shown to be isomorphic to the cotangent space $ T_{x}^{*}M$ through the use of Taylor's theorem. The tangent space $T_{x}M$ may then be defined as the dual space of $I/I^{2}$. While this definition is the most abstract, it is also the one that is most easily transferable to other settings, for instance, to the varieties considered in algebraic geometry.
Thanks!
