So I have the following definition for a Rimannian metric:
Let $M$ be a smooth manifold, $g$ a $(0,2)$ tensorfield on $M$, s.t. $g_p$ is a scalar product on $T_pM$ and $g_p$ is positive. Now, a $(0,2)$ tensor field is a $C^{\infty}(M)$-linear map
$\mathcal{X}(M) \times \mathcal{X}(M) \rightarrow C^{\infty}(M)$ where $\mathcal{X}(M)$ is the space of all vector fields on $M$, functions.
Now we didn't really get an explanation what $g_p$ is supposed to be.
$g_p$ has to be a bilinear function $T_pM \times T_pM \rightarrow \mathbb{R}$.
Where do I have to put $p$ in, s.t. a function from $\mathcal{X}(M) \times \mathcal{X}(M) \rightarrow C^{\infty}(M)$ becomes a function from $T_pM \times T_pM \rightarrow \mathbb{R}$?
I'm sorry if this is a stupid question but I really don't get it here..
If $X,Y\in \mathfrak{X}(M) $ then $g(X,Y)\in C^\infty(M)$ and you can evaluate it at a point. By definition \begin{equation} g(X,Y)|_p=g_p\,(X_p,Y_p). \end{equation} It may help to recall that $\mathfrak{X}(M)=\Gamma(TM)$, the space of smooth section of the tangent bundle, and $TM=\bigcup_{p\in M}\ \ T_p M$.