Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some map $(U,\varphi)$. Another characterization of this condition is $(f\circ\gamma_1)'(0)=(f\circ\gamma_2)'(0)$ for every real valued funcion from $C^k(M)$.

Usually, vector space structure of T is define in some map U, p $\in U$. My question is follows. Can I define vector space structure of T without map? I mean, in terms of real valued functions?

Let $\gamma$ be curve, $t\mapsto\gamma(t)$. If $[\gamma]$ is class from T, then I can define scalar multiplication $c[\gamma]$, like $c\gamma:t\mapsto\gamma(ct)$. But I do not know how can I define addition.

• I didnn't understand the "without map" part, but you can define via derivations and then applies localization. – user40276 Feb 2 '14 at 15:23
• Now, I want to concentrato to curves. Derivations are clear. – user109301 Feb 2 '14 at 15:32
• If you want to use real valued functions you'll need the sheaf over $M$. And about the addition you'll have to specify the point, because $T$ is a vector bundle, not a vector space. – user40276 Feb 2 '14 at 15:38
• If I want to define vector sturcture to T, I need map $(U,g)$ to define scalar multiplication and vec add. If a and b are curve classes and p,q are reals, then I define linear combination pa+qb to be a curve $c(t)=g^{-1}(p*g\circ a(t)+q*g\circ b(t)+(1-p-q)g(x))$, that is vector space structure of T in point x. – user109301 Feb 2 '14 at 15:40