I think this is the dual question to my previous question.
From Gudmundsoon notes, page 60:
We can model the hyperbolic space $\mathbb H^m$ as the super half plane space $\mathbb R^+ \times \mathbb R^{m-1}$ with the metric
$$g_p(X, Y) = \frac{1}{x_1^2} \langle X, Y \rangle_{\mathbb R^m}, $$
where $x = (x_1, \dots, x_m) \in H^m$.
My question is: what is the relationship between the metric and the inner product? What is the "right" way of multiplying those vectors in a space that has this curvature? Clearly the "flat" one inherited from $\mathbb R ^m$ won't make much sense here.
Follow up question: more generally, what's the right way of constructing the inner product on $(\mathbb R^ m, g)$ for an arbitrary metric?
You seem to be missing some key concepts regarding tangent spaces and Riemannian metrics.
In general, a Riemannian metric on a manifold $M$ is a family of inner products, one for each point $p \in M$.
Given $p,q \in M$, the tangent spaces at $p$ and $q$ are disjoint vector spaces $T_p M$ and $T_q M$. The disjoint union of all these vector spaces is called the tangent bundle and is denoted $$TM = \coprod_{p \in M} T_p M $$
What a Riemannian metric does is to assign an inner product on each $T_p M$. There is not just one inner product.
In the example of $\mathbb H^n$, $T \mathbb H^n = \mathbb H^n \times \mathbb R^n$ and $T_p \mathbb H^n = \{p\} \times \mathbb R^n$. The vector space structure on $T_p \mathbb H^n$ is $(p,v) + (p,w) = (p,v+w)$, and $r (p,v) = (p,rv)$. Given $p = (x_1,...,x_n) \in \mathbb H^n$, and given $\vec v,\vec w \in \mathbb R^n$, what $g_p$ does is to assign the inner product $$\langle (p,\vec v) , (p,\vec w) \rangle = \frac{\langle \vec v, \vec w \rangle_{\mathbb R^n}}{x_1^2} $$