1. Bijection between tangent spaces

Hello,

We have defined the tangent space at a point p of a manifold as the set of all derivatives.
Now i want to show, that there exist a bijection between all derivatives and the "Tangentvectors" defined by (equivalent classes of) paths.
More precisely:
We call two paths equivalent, iff they define the same derivation at a point p. Now i want to show that there is a bijection between the set of all equivalent classes and the Tangentspace (defined as the set of derivatives).

My Approach was the following.
Every path c:I->M into the manifold M defines a derivation by
$\displaystyle D(f):=\frac{d}{dt}_{|t=0}(f\circ c)(t)$

so i try to show that $\displaystyle \phi([c])=D$ with $\displaystyle D(f)=\frac{d}{dt}_{|t=0}(f\circ c)(t)$ is a bijection.

Of course $\displaystyle \phi$ is well defined and injective, but i couldn't show that
$\displaystyle \phi$ is surjective!

If D is some arbitrary derivative at the point p. How can i find a path c, s.t. $\displaystyle \phi(c)=D$?

Regards

2. If you fix a point $\displaystyle p$ and express everything in a local chart, then the required path $\displaystyle c(s), \ -\epsilon<s<\epsilon$ is just the unique solution of the initial value problem $\displaystyle c'(0)=D, \ c(0)=p$.

3. Hello Rebesques,

thank you for your help! But i don't see what you mean. Ok lets choose a chart (U,$\displaystyle \phi$) around the fixed point p.
I think by your notation you mean $\displaystyle c'(0)=\frac{d}{dt}_{|t=0}(f\circ c)(t)$, is it right?

But How can i solve this differential equation D=c'(t)?

But i know by a thm., that every derivation D is of the form
$\displaystyle D=\sum_{i=1}^d D(x_i) \frac{d}{dx_i}_{|p}$

whereas $\displaystyle x_i$ are the coordinate functions of our selected chart $\displaystyle \phi$
Perhaps this equation can help? But i don't know how.

4. Not quite. We are looking for a curve $\displaystyle c-\epsilon,\epsilon)\rightarrow M$ with $\displaystyle c(0)=p, \ c'(0)=(\frac{\partial}{\partial t}\vert_{0})c=D\in T_pM$. There are many curves with this property; Let's construct one.

Choose a chart $\displaystyle (U,\phi), \ \phi=(x^1,\ldots,x^n)$, and w.l.o.g. assume $\displaystyle \phi(p)=0, \ D(\phi)\vert_{p}=I$, where $\displaystyle D(\phi)\vert_{p}=(\frac{\partial}{\partial x^i}\vert_p(x^j))$ and I is the identity matrix.

Then, by letting $\displaystyle D=a^i\frac{\partial}{\partial x^i}$ and identifying $\displaystyle D=(a^1,\ldots,a^n)\in R^n$, the curve $\displaystyle c(t)=\phi^{-1}(tD), \ -\epsilon<t<\epsilon$ has the required properties.

Ps. And $\displaystyle c$ is unique in the sence it belongs to the equivalence class of curves passing through $\displaystyle p$ and having tangent $\displaystyle D\in T_pM$.