Herman Karcher says at the end of the first page in Riemannian Center of Mass and so called karcher mean that for the Hiperbolic space $\mathbb{H}^n$ and a hemisphere of $\mathbb{S}^n$ there are standard isometrics embeddings into $\mathbb{R}^{n+1}$.
I would like to know if this means that there exist a transformation $\Phi$ such that for $d(x,y) = \| \Phi(x) - \Phi(y) \|_2$. If this is true, then which is such transformation in the case of a hemisphere of $\mathbb{S}^n$ with the great circle distance i.e. $d(x,y) = \arccos(\langle x, y \rangle)$
Actually he just says that "The hyperbolic space $\mathbb{H}^n$ and a hemisphere of $\mathbb{S}^n$ are not linear spaces, but they have standard isometric embeddings into $\mathbb{R}^{n+1}$". Your citation is slightly confusing as it seems to imply that the distance between the two manifolds is preserved, which does not make sense.
What it means is that they can be embedded into (Euclidean or Minkowski) $n+1$ space such that the induced distance is the same as the distance in the abstract Riemannian manifolds, i.e. the distance between any two points $x, y$ is the $\inf$ of all lengths of curves (in the manifold) joining $x$ and $y$.
The embedding of the sphere is just the boundary of the unit ball. The embedding of hyperbolic space is (one component) of the unit sphere with respect to the Minkowski Metric (i.e the set defined by $x_0^2 = \sum_{i>0} x_i^2$ (and is isometric wrt to the Minkowski metric).