Question 1: Does the metric define the inner product or does the inner product define the metric? I have $$(u,v) := \delta_{ij} u^i v^j$$ and I don't know if the $\delta$ is the metric or just the delta function. In the definition of the metric tensor one also uses the inner product but in that case it is always $$(u,v) := u^i v^i$$ So I guess the metric defines the inner product. Is that correct?
Question 2: In the definition of an isometry, $\phi$ we have $$(u,v)_{g_1} = (\phi(u), \phi(v))_{g_2}$$ What does this notation mean? Simpily $$ g_{1_{ij}} u^i v^i = g_{2_{ij}} \phi(u)^i \phi(u)^j $$ or something else?
If $c:(I,|\ |)\rightarrow (X,d)$ is a continous curve in metric space, then define a length of $c$ to be $$ \lim_n\ \sum_{i=1}^n\ d(c(t_i),c(t_{i+1})) $$ where $t_i$ is a partition on $I$.
Here if $p=c(0),\ q=c(1)$, then infimum of lengths of curves between $p$ and $q$ is intrinsic metric. One of example of intrinsic metric is norm on $\mathbb{R}^n$. Here if norm satisfies $\| v-w\|^2 + \|v+w\|^2 =2(\|v\|^2+\|w\|^2)$, then the norm is in fact inner product $g$. Further, in inner product space we can define a length of curve as followes : $$ {\rm length}\ c =\int_0^1\ g(c'(t),c'(t))^\frac{1}{2}\ dt \ \ast$$
Consider inner product space $(X,g)$. From $\ast$, we have a metric $d_g$ which is a infimum of length of curves. If $f :(X,g)\rightarrow (X',g')$ is bijective isometry, then $f$ is isometry between $d_g,\ d_{g'}$, i.e. $$ d_g(x,y)=d_{g'}(f(x),f(y))$$