I am reading the Riemannian metric and I have some doubts so first let me write everything that I have understood, correct me if I am wrong, thanks
Let $S$ be a manifold. For each point $p\in S$ let us define a map $g:p \mapsto \langle \; , \; \rangle_p$ where $\langle \; , \; \rangle: T_p(S)\times T_p(S)\to \mathbb{R}$ and the inner product satisfies the conditions such as (1)Bilinear,(2) Symmetry and (3) Positive-definiteness,we call this a $\textbf{Riemannian Metric}$ on $S$. and $(S,g)$ is known as $\textbf{Riemannian manifold.}$
so basically we have a map $g$ which maps each point of $S$ to an element of covariant two tensors.
Now let $[\xi^i]$ be a coordinate system for $S$ and $\partial_i=\frac{\partial}{\partial\xi^i}$then the components $\{g_{ij};i,j=1,2,..,n\}(n=dimS)$ of Riemannian metric $g$ with respect to $[\xi^i]$ are determined by $g_{ij}=\langle (\partial_i)\;,\;(\partial_j) \rangle$ .This is a $C^{\infty}$ function that maps each point $p\in S$ to $g_{ij}(p)=\langle (\partial_i)_p\;,\;(\partial_j)_p \rangle_p$ and let suppose we have $D,D'\in T_p({S})$ then in terms of component we can write it as $D=D^i(\partial_i)_p$ and $D'=D'^{i}(\partial_i)_p$ and the inner product is given by $\langle D\;,\;D' \rangle_p=g_{ij}D^{i}D'^{j}$.
Now I am trying to construct an example for this by taking $S=\mathbb{R^2}$ with the coordinates $\{\xi^1,\xi^2\}$ and we will have $T_{p}\mathbb{R^2}\simeq\mathbb{R^2}$ and then since $(\partial_i)_p$ is the basis of the $T_{p}\mathbb{R^2}$ so simply I have assumed the standard basis $(\partial_1)p=(1,0)$ and $(\partial_2)_p=(0,1)$ and then if we calculate $g_{ij}(p)$ which is given by $g_{ij}(p)=\langle (\partial_i)_p\;,\;(\partial_j)_p \rangle$ then we can calculate all the coefficient and then if we choose any $D,D'\in T_{p}R^2$ such that $D=(2,3)$ and $D'=(2,4)$ then we can calculate the inner product using the above relation of inner product of $D \text{ and } D'$
My question is what is the role of $p$ here since once I defined the basis then at every point I am going to take the same basis throughout and calculate $g_{ij}$ if I change the basis say $(\partial_1)p=(1,2)$ and $(\partial_2)_p=(2,1)$ then also at every point I am going to take this same basis then what is the role of $p$?? can someone explain it what am I missing,That will be great help.
