Let $S^n$ be the unit n-sphere embedded in $\mathbb{R}^{n+1}$:
$$ S^n = \{ a \in \mathbb{R}^{n+1} \mid a \cdot a = 1 \} $$
What is the induced metric tensor for the sphere, in $\mathbb{R}^{n+1}$ coordinates?
$$ g(x_1, x_2, \ldots, x_n, x_{n+1})$$
Note that
$$ \text{arclength}(a,b) = \arccos(a \cdot b) $$
How do I obtain the metric tensor components from this expression? Should I take $b=a+\delta a$?
The induced metric is just the Euclidean metric on $\mathbb R^{n+1}$, i.e. $g_{ij} = \delta_{ij}$, but restricted to act on vectors tangent to $S^n$. You need to choose a system of $n$ coordinates parametrizing $S^n$ alone (or an $n$-frame tangent to $S^n$) if you want to get an $n \times n$ matrix of components for $g$.